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aria-label="群"><!--[--><!--[--><!--]--><!--]-->群<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_1.html" aria-label="数论（一）"><!--[--><!--[--><!--]--><!--]-->数论（一）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_2.html" aria-label="数论（二）"><!--[--><!--[--><!--]--><!--]-->数论（二）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_3.html" aria-label="数论（三）"><!--[--><!--[--><!--]--><!--]-->数论（三）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/probability.html" aria-label="概率"><!--[--><!--[--><!--]--><!--]-->概率<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">密码学 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/rsa.html" aria-label="RSA"><!--[--><!--[--><!--]--><!--]-->RSA<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/flip_coin.html" aria-label="抛币协议"><!--[--><!--[--><!--]--><!--]-->抛币协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/mental_poker.html" aria-label="智能扑克协议"><!--[--><!--[--><!--]--><!--]-->智能扑克协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item active collapsible">图形学 <span class="down arrow"></span></p><ul style="" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item active collapsible">数学基础 <span class="down arrow"></span></p><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link route-link-active auto-link vp-sidebar-item active" href="/note/graphics/math/math_01.html" aria-label="矢量"><!--[--><!--[--><!--]--><!--]-->矢量<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_1-基础约定" aria-label="1. 基础约定"><!--[--><!--[--><!--]--><!--]-->1. 基础约定<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_1-1-坐标系" aria-label="1.1 坐标系"><!--[--><!--[--><!--]--><!--]-->1.1 坐标系<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_1-2-左乘和右乘" aria-label="1.2 左乘和右乘"><!--[--><!--[--><!--]--><!--]-->1.2 左乘和右乘<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_1-3-行存储和列存储" aria-label="1.3 行存储和列存储"><!--[--><!--[--><!--]--><!--]-->1.3 行存储和列存储<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="#_2-矢量" aria-label="2. 矢量"><!--[--><!--[--><!--]--><!--]-->2. 矢量<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_2-1-基本操作" aria-label="2.1 基本操作"><!--[--><!--[--><!--]--><!--]-->2.1 基本操作<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_2-2-标准矢量" aria-label="2.2 标准矢量"><!--[--><!--[--><!--]--><!--]-->2.2 标准矢量<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_2-3-长度" aria-label="2.3 长度"><!--[--><!--[--><!--]--><!--]-->2.3 长度<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_2-4-点积" aria-label="2.4 点积"><!--[--><!--[--><!--]--><!--]-->2.4 点积<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_2-5叉积" aria-label="2.5叉积"><!--[--><!--[--><!--]--><!--]-->2.5叉积<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_2-6-混合积" aria-label="2.6 混合积"><!--[--><!--[--><!--]--><!--]-->2.6 混合积<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_2-7-三重积" aria-label="2.7 三重积"><!--[--><!--[--><!--]--><!--]-->2.7 三重积<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_02.html" aria-label="矩阵"><!--[--><!--[--><!--]--><!--]-->矩阵<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_03.html" aria-label="立体角"><!--[--><!--[--><!--]--><!--]-->立体角<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_01.html" aria-label="几何变换(一)"><!--[--><!--[--><!--]--><!--]-->几何变换(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_02.html" aria-label="几何变换(二)"><!--[--><!--[--><!--]--><!--]-->几何变换(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_03.html" aria-label="法线变换"><!--[--><!--[--><!--]--><!--]-->法线变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_04.html" aria-label="摄像机变换"><!--[--><!--[--><!--]--><!--]-->摄像机变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">光照模型 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/classic/illumination_model_01.html" aria-label="传统光照模型"><!--[--><!--[--><!--]--><!--]-->传统光照模型<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/luminosity/luminosity.html" aria-label="光度学"><!--[--><!--[--><!--]--><!--]-->光度学<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/BRDF/BRDF.html" aria-label="双向反射分布函数(BRDF)"><!--[--><!--[--><!--]--><!--]-->双向反射分布函数(BRDF)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_1.html" aria-label="微平面理论(一)"><!--[--><!--[--><!--]--><!--]-->微平面理论(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_2.html" aria-label="微平面理论(二)"><!--[--><!--[--><!--]--><!--]-->微平面理论(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_3.html" aria-label="微平面理论(三)"><!--[--><!--[--><!--]--><!--]-->微平面理论(三)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/render_function/render_function_1.html" aria-label="光照方程"><!--[--><!--[--><!--]--><!--]-->光照方程<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">环境光渲染 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_01.html" aria-label="环境光渲染(一)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_02.html" aria-label="环境光渲染(二)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">编程语言 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">JavaScript <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/helloworld.html" aria-label="环境搭建"><!--[--><!--[--><!--]--><!--]-->环境搭建<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/grammar.html" aria-label="基本语法"><!--[--><!--[--><!--]--><!--]-->基本语法<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/function.html" aria-label="函数"><!--[--><!--[--><!--]--><!--]-->函数<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/object_and_class.html" aria-label="对象和类"><!--[--><!--[--><!--]--><!--]-->对象和类<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul><!--[--><!--]--></aside><!--]--><!--[--><main class="vp-page"><!--[--><!--]--><div vp-content><!--[--><!--]--><div><h1 id="数学基础-一" tabindex="-1"><a class="header-anchor" href="#数学基础-一"><span>数学基础(一）</span></a></h1><hr><h2 id="_1-基础约定" tabindex="-1"><a class="header-anchor" href="#_1-基础约定"><span>1. 基础约定</span></a></h2><h3 id="_1-1-坐标系" tabindex="-1"><a class="header-anchor" href="#_1-1-坐标系"><span>1.1 坐标系</span></a></h3><h4 id="_1-1-1-左手坐标系" tabindex="-1"><a class="header-anchor" href="#_1-1-1-左手坐标系"><span>1.1.1 左手坐标系</span></a></h4><figure><img src="/assets/axis_left-D307nuvU.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>DirectX一般缺省使用左手坐标系</p><h4 id="_1-1-2-右手坐标系" tabindex="-1"><a class="header-anchor" href="#_1-1-2-右手坐标系"><span>1.1.2 右手坐标系</span></a></h4><figure><img src="/assets/axis_right-DlLDxrsw.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>OpenGL缺省使用右手坐标系，图形学中缺省使用这种坐标系</p><h3 id="_1-2-左乘和右乘" tabindex="-1"><a class="header-anchor" href="#_1-2-左乘和右乘"><span>1.2 左乘和右乘</span></a></h3><h4 id="_1-2-1-左乘-前乘-pre-multiply" tabindex="-1"><a class="header-anchor" href="#_1-2-1-左乘-前乘-pre-multiply"><span>1.2.1 左乘，前乘(pre-multiply)</span></a></h4><p>矢量为行向量，和矩阵的乘法为</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 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class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>12</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>13</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>21</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>22</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>23</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>31</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>32</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>33</mn></mrow></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><p>DirectX缺省使用左乘</p><h4 id="_1-2-2-右乘-后乘-post-multiply" tabindex="-1"><a class="header-anchor" href="#_1-2-2-右乘-后乘-post-multiply"><span>1.2.2 右乘，后乘(post-multiply)</span></a></h4><p>矢量为列向量，和矩阵的乘法为</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mrow><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:7.28em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><mjx-mrow space="2"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:0.572em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-bottom:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;padding-bottom:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>12</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>13</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>21</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>22</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>23</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>31</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>32</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>33</mn></mrow></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><p>OpenGL缺省使用右乘，图形学中缺省使用这种方式。</p><h3 id="_1-3-行存储和列存储" tabindex="-1"><a class="header-anchor" href="#_1-3-行存储和列存储"><span>1.3 行存储和列存储</span></a></h3><h4 id="_1-3-1-行存储" tabindex="-1"><a class="header-anchor" href="#_1-3-1-行存储"><span>1.3.1 行存储</span></a></h4><p>在计算机内存中，按照行优先存储一个矩阵，DirectX使用这种方式 比如对于矩阵</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mrow><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:7.28em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>12</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>13</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>21</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>22</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>23</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>31</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>32</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>33</mn></mrow></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><p>在内存中为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2026"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub><mo>,</mo><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>12</mn></mrow></msub><mo>,</mo><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>13</mn></mrow></msub><mo>,</mo><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>21</mn></mrow></msub><mo>,</mo><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>22</mn></mrow></msub><mo>…</mo></math></mjx-assistive-mml></mjx-container></p><h4 id="_1-3-2-列存储" tabindex="-1"><a class="header-anchor" href="#_1-3-2-列存储"><span>1.3.2 列存储</span></a></h4><p>按照列优先存储一个矩阵，OpenGL使用这种方式 比如对于矩阵</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mrow><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:7.28em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>12</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>13</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>21</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>22</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>23</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>31</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>32</mn></mrow></msub></mtd><mtd><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>33</mn></mrow></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><p>在内存中为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2026"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub><mo>,</mo><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>21</mn></mrow></msub><mo>,</mo><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>31</mn></mrow></msub><mo>,</mo><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>12</mn></mrow></msub><mo>,</mo><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mn>22</mn></mrow></msub><mo>…</mo></math></mjx-assistive-mml></mjx-container></p><h2 id="_2-矢量" tabindex="-1"><a class="header-anchor" href="#_2-矢量"><span>2. 矢量</span></a></h2><h3 id="_2-1-基本操作" tabindex="-1"><a class="header-anchor" href="#_2-1-基本操作"><span>2.1 基本操作</span></a></h3><h4 id="_2-1-1-定义" tabindex="-1"><a class="header-anchor" href="#_2-1-1-定义"><span>2.1.1 定义</span></a></h4><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:15.757em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.27em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cB1"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.27em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cB1"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cB1"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cB1"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">[</mo><msub><mi>u</mi><mi>x</mi></msub><mo>,</mo><msub><mi>u</mi><mi>y</mi></msub><mo>,</mo><msub><mi>u</mi><mi>z</mi></msub><msup><mo stretchy="false">]</mo><mi>T</mi></msup><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mo stretchy="false">[</mo><msub><mi>v</mi><mi>x</mi></msub><mo>,</mo><msub><mi>v</mi><mi>y</mi></msub><mo>,</mo><msub><mi>v</mi><mi>z</mi></msub><msup><mo stretchy="false">]</mo><mi>T</mi></msup></mtd></mtr><mtr><mtd><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo>±</mo><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">→</mo></mover></mrow></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">[</mo><msub><mi>u</mi><mi>x</mi></msub><mo>±</mo><msub><mi>v</mi><mi>x</mi></msub><mo>,</mo><msub><mi>u</mi><mi>y</mi></msub><mo>±</mo><msub><mi>v</mi><mi>y</mi></msub><mo>,</mo><msub><mi>u</mi><mi>z</mi></msub><mo>±</mo><msub><mi>v</mi><mi>z</mi></msub><msup><mo stretchy="false">]</mo><mi>T</mi></msup></mtd></mtr><mtr><mtd><mi>k</mi><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">[</mo><mi>k</mi><msub><mi>u</mi><mi>x</mi></msub><mo>,</mo><mi>k</mi><msub><mi>u</mi><mi>y</mi></msub><mo>,</mo><mi>k</mi><msub><mi>u</mi><mi>z</mi></msub><msup><mo stretchy="false">]</mo><mi>T</mi></msup></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><h4 id="_2-1-2-几何意义" tabindex="-1"><a class="header-anchor" href="#_2-1-2-几何意义"><span>2.1.2 几何意义</span></a></h4><figure><img src="/assets/vector_op-EGHzijwY.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><h3 id="_2-2-标准矢量" tabindex="-1"><a class="header-anchor" href="#_2-2-标准矢量"><span>2.2 标准矢量</span></a></h3><figure><img src="/assets/standard_basis-DLyK0dsc.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>标准矢量(Standard Basis)，定义<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.203em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D48A TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.235em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D48B TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.302em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D48C TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">i</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">j</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">k</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup></math></mjx-assistive-mml></mjx-container></p><h3 id="_2-3-长度" tabindex="-1"><a class="header-anchor" href="#_2-3-长度"><span>2.3 长度</span></a></h3><h4 id="_2-3-1-定义" tabindex="-1"><a class="header-anchor" href="#_2-3-1-定义"><span>2.3.1 定义</span></a></h4><p>对于三维矢量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>，定义矢量的长度</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msqrt space="4"><mjx-sqrt><mjx-surd><mjx-mo class="mjx-lop"><mjx-c class="mjx-c221A TEX-S2"></mjx-c></mjx-mo></mjx-surd><mjx-box style="padding-top:0.379em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-spacer style="margin-top:0.223em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msubsup space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-spacer style="margin-top:0.223em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msubsup space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-spacer style="margin-top:0.223em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup></mjx-box></mjx-sqrt></mjx-msqrt></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo data-mjx-texclass="ORD">∥</mo><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD">∥</mo><mo>=</mo><msqrt><msubsup><mi>u</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>u</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>u</mi><mi>z</mi><mn>2</mn></msubsup></msqrt></math></mjx-assistive-mml></mjx-container><p>有时候矢量的长度也可以用<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD" stretchy="false">|</mo></math></mjx-assistive-mml></mjx-container>表示，但容易和绝对值混淆</p><h4 id="_2-3-2-定义" tabindex="-1"><a class="header-anchor" href="#_2-3-2-定义"><span>2.3.2 定义</span></a></h4><p>对于三维矢量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>，定义矢量的单位矢量</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0;padding-left:0.314em;margin-bottom:-0.531em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c5E"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mfrac space="2"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mfrac space="2"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mover><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">^</mo></mover></mrow><mo>=</mo><mfrac><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mrow><mo data-mjx-texclass="ORD">∥</mo><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD">∥</mo></mrow></mfrac><mo>=</mo><mo stretchy="false">(</mo><mfrac><msub><mi>u</mi><mi>x</mi></msub><mrow><mo data-mjx-texclass="ORD">∥</mo><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD">∥</mo></mrow></mfrac><mo>,</mo><mfrac><msub><mi>u</mi><mi>y</mi></msub><mrow><mo data-mjx-texclass="ORD">∥</mo><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD">∥</mo></mrow></mfrac><mo>,</mo><mfrac><msub><mi>u</mi><mi>z</mi></msub><mrow><mo data-mjx-texclass="ORD">∥</mo><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD">∥</mo></mrow></mfrac><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><h3 id="_2-4-点积" tabindex="-1"><a class="header-anchor" href="#_2-4-点积"><span>2.4 点积</span></a></h3><h4 id="_2-4-1-定义" tabindex="-1"><a class="header-anchor" href="#_2-4-1-定义"><span>2.4.1 定义</span></a></h4><p>对于两个三维矢量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.27em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>，定义它们的点积(<a href="https://en.wikipedia.org/wiki/Dot_product" target="_blank" rel="noopener noreferrer">Dot Product</a>)为</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.27em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><msub><mi>u</mi><mi>x</mi></msub><msub><mi>v</mi><mi>x</mi></msub><mo>+</mo><msub><mi>u</mi><mi>y</mi></msub><msub><mi>v</mi><mi>y</mi></msub><mo>+</mo><msub><mi>u</mi><mi>z</mi></msub><msub><mi>v</mi><mi>z</mi></msub></math></mjx-assistive-mml></mjx-container><h4 id="_2-4-2-点积的几何意义" tabindex="-1"><a class="header-anchor" href="#_2-4-2-点积的几何意义"><span>2.4.2 点积的几何意义</span></a></h4><figure><img src="/assets/dot_product-LEv427NC.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mo data-mjx-texclass="ORD">∥</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD">∥</mo><mo data-mjx-texclass="ORD">∥</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD">∥</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mi>θ</mi></mrow></math></mjx-assistive-mml></mjx-container><p>两个向量的点积可以看成两个向量相近的程度，向量和自身的点积是向量长度的平方</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msubsup space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-spacer style="margin-top:0.347em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msubsup space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-spacer style="margin-top:0.347em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mo data-mjx-texclass="ORD">∥</mo><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD">∥</mo><mo data-mjx-texclass="ORD">∥</mo><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD">∥</mo><mo>=</mo><msubsup><mi>u</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>u</mi><mi>y</mi><mn>2</mn></msubsup></math></mjx-assistive-mml></mjx-container><p>向量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>在向量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>上的投影的长度为</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>a</mi><mi>b</mi></msub><mo>=</mo><mfrac><mrow><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow></mrow><mrow><mo data-mjx-texclass="ORD">∥</mo><mi>b</mi><mo data-mjx-texclass="ORD">∥</mo></mrow></mfrac></math></mjx-assistive-mml></mjx-container><h4 id="_2-4-3-推论" tabindex="-1"><a class="header-anchor" href="#_2-4-3-推论"><span>2.4.3 推论</span></a></h4><p>判断两个矢量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.372em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D496 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.315em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D497 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">u</mi><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">v</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>之间的夹角和它们之间点积的关系</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.372em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D496 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.315em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D497 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.372em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D496 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.315em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D497 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">u</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">v</mi><mo stretchy="false">→</mo></mover></mrow></mrow><mrow><mo data-mjx-texclass="ORD">∥</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">u</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD">∥</mo><mo data-mjx-texclass="ORD">∥</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">v</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD">∥</mo></mrow></mfrac></math></mjx-assistive-mml></mjx-container><ul><li>当<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.27em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3E"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">→</mo></mover></mrow><mo>></mo><mn>0</mn></math></mjx-assistive-mml></mjx-container>，夹角在[0,90)之间</li><li>当<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.27em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container>，两个矢量垂直</li><li>当<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.27em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">→</mo></mover></mrow><mo><</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container>，两个矢量夹角在(90,180]之间</li></ul><h4 id="_2-4-3-基本运算规则" tabindex="-1"><a class="header-anchor" href="#_2-4-3-基本运算规则"><span>2.4.3 基本运算规则</span></a></h4><p>矢量的点积满足一般的交换律、分配律</p><ul><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container></li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container></li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>k</mi><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mo stretchy="false">(</mo><mi>k</mi><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>k</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></li></ul><h3 id="_2-5叉积" tabindex="-1"><a class="header-anchor" href="#_2-5叉积"><span>2.5叉积</span></a></h3><h4 id="_2-5-1-定义" tabindex="-1"><a class="header-anchor" href="#_2-5-1-定义"><span>2.5.1 定义</span></a></h4><p>对于两个三维矢量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>，定义它们的叉积</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:24.699em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.317em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D482 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.261em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D483 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:4.237em;vertical-align:-1.868em;" class="mjx-c7C"><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:4.916em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.203em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D48A TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.235em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D48B TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.302em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D48C TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:4.237em;vertical-align:-1.868em;" class="mjx-c7C"><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.203em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D48A TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.235em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D48B TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.302em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D48C TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">b</mi><mo stretchy="false">→</mo></mover></mrow></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">i</mi><mo stretchy="false">→</mo></mover></mrow></mtd><mtd><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">j</mi><mo stretchy="false">→</mo></mover></mrow></mtd><mtd><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">k</mi><mo stretchy="false">→</mo></mover></mrow></mtd></mtr><mtr><mtd><msub><mi>a</mi><mi>x</mi></msub></mtd><mtd><msub><mi>a</mi><mi>y</mi></msub></mtd><mtd><msub><mi>a</mi><mi>z</mi></msub></mtd></mtr><mtr><mtd><msub><mi>b</mi><mi>x</mi></msub></mtd><mtd><msub><mi>b</mi><mi>y</mi></msub></mtd><mtd><msub><mi>b</mi><mi>z</mi></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">|</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mi>y</mi></msub><msub><mi>b</mi><mi>z</mi></msub><mo>−</mo><msub><mi>a</mi><mi>z</mi></msub><msub><mi>b</mi><mi>y</mi></msub><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">i</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mi>z</mi></msub><msub><mi>b</mi><mi>x</mi></msub><mo>−</mo><msub><mi>a</mi><mi>x</mi></msub><msub><mi>b</mi><mi>z</mi></msub><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">j</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mi>x</mi></msub><msub><mi>b</mi><mi>y</mi></msub><mo>−</mo><msub><mi>a</mi><mi>y</mi></msub><msub><mi>b</mi><mi>x</mi></msub><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">k</mi><mo stretchy="false">→</mo></mover></mrow></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><h4 id="_2-5-2-叉积的几何意义" tabindex="-1"><a class="header-anchor" href="#_2-5-2-叉积的几何意义"><span>2.5.2 叉积的几何意义</span></a></h4><p>在三维几何中，向量a和向量b的叉乘结果是一个向量，更为熟知的叫法是法向量，该向量垂直于a和b向量构成的平面。</p><figure><img src="/assets/cross_product-2IEbK7Oi.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>定义向量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.356em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D48F TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">n</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>是一个单位向量，方向同时垂直于向量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.372em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D496 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">u</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>和<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.315em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D497 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">v</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>，且符合右手定则（右手坐标系），那么</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.372em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D496 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.315em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D497 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.372em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D496 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.315em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D497 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6E"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.356em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D48F TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">u</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">v</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mo data-mjx-texclass="ORD">∥</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">u</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD">∥</mo><mo data-mjx-texclass="ORD">∥</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">v</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD">∥</mo><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">n</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><p>向量a和向量b的叉积还可以视作以<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>和<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>为边的平行四边形的面积</p><h4 id="_2-5-3-基本运算规则" tabindex="-1"><a class="header-anchor" href="#_2-5-3-基本运算规则"><span>2.5.3 基本运算规则</span></a></h4><ul><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container></li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.25em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mover><mn>0</mn><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container></li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container></li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mo stretchy="false">(</mo><mi>k</mi><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>k</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></li></ul><h4 id="_2-5-4-叉积的矩阵形式" tabindex="-1"><a class="header-anchor" href="#_2-5-4-叉积的矩阵形式"><span>2.5.4 叉积的矩阵形式</span></a></h4><p>两个向量的叉积，可以表达成一个矩阵和向量的乘积的形式，例如向量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>和<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>的叉积</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c 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class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.89em;vertical-align:-1.695em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:7.25em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.89em;vertical-align:-1.695em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><mjx-mrow space="2"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.845em;vertical-align:-1.672em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:0.916em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.845em;vertical-align:-1.672em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow></mrow><msub><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mi>x</mi></mrow></msub></mtd></mtr><mtr><mtd><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow></mrow><msub><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mi>y</mi></mrow></msub></mtd></mtr><mtr><mtd><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow></mrow><msub><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mi>z</mi></mrow></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>a</mi><mi>y</mi></msub><msub><mi>b</mi><mi>z</mi></msub><mo>−</mo><msub><mi>a</mi><mi>z</mi></msub><msub><mi>b</mi><mi>y</mi></msub></mtd></mtr><mtr><mtd><msub><mi>a</mi><mi>z</mi></msub><msub><mi>b</mi><mi>x</mi></msub><mo>−</mo><msub><mi>a</mi><mi>x</mi></msub><msub><mi>b</mi><mi>z</mi></msub></mtd></mtr><mtr><mtd><msub><mi>a</mi><mi>x</mi></msub><msub><mi>b</mi><mi>y</mi></msub><mo>−</mo><msub><mi>a</mi><mi>y</mi></msub><msub><mi>b</mi><mi>x</mi></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>0</mn></mtd><mtd><mo>−</mo><msub><mi>a</mi><mi>z</mi></msub></mtd><mtd><msub><mi>a</mi><mi>y</mi></msub></mtd></mtr><mtr><mtd><msub><mi>a</mi><mi>z</mi></msub></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><msub><mi>a</mi><mi>x</mi></msub></mtd></mtr><mtr><mtd><mo>−</mo><msub><mi>a</mi><mi>y</mi></msub></mtd><mtd><msub><mi>a</mi><mi>x</mi></msub></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>b</mi><mi>x</mi></msub></mtd></mtr><mtr><mtd><msub><mi>b</mi><mi>y</mi></msub></mtd></mtr><mtr><mtd><msub><mi>b</mi><mi>z</mi></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><p>这里称这个矩阵为矢量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>的叉乘矩阵<a href="https://en.wikipedia.org/wiki/Cross_product#Conversion_to_matrix_multiplication" target="_blank" rel="noopener noreferrer">cross-product matrix</a>，写作<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D482 TEX-BI"></mjx-c></mjx-mi><mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi mathvariant="bold-italic">a</mi><msub><mo stretchy="false">]</mo><mrow data-mjx-texclass="ORD"><mo>×</mo></mrow></msub></math></mjx-assistive-mml></mjx-container></p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D482 TEX-BI"></mjx-c></mjx-mi><mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mover space="4"><mjx-over style="padding-bottom:0.111em;"><mjx-mrow size="s"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-over><mjx-base style="padding-left:0.154em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo></mjx-base></mjx-mover><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.89em;vertical-align:-1.695em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:7.25em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.89em;vertical-align:-1.695em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">[</mo><mi mathvariant="bold-italic">a</mi><msub><mo stretchy="false">]</mo><mrow data-mjx-texclass="ORD"><mo>×</mo></mrow></msub><mover><mo>=</mo><mrow><mi>d</mi><mi>e</mi><mi>f</mi></mrow></mover><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>0</mn></mtd><mtd><mo>−</mo><msub><mi>a</mi><mi>z</mi></msub></mtd><mtd><msub><mi>a</mi><mi>y</mi></msub></mtd></mtr><mtr><mtd><msub><mi>a</mi><mi>z</mi></msub></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><msub><mi>a</mi><mi>x</mi></msub></mtd></mtr><mtr><mtd><mo>−</mo><msub><mi>a</mi><mi>y</mi></msub></mtd><mtd><msub><mi>a</mi><mi>x</mi></msub></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><h3 id="_2-6-混合积" tabindex="-1"><a class="header-anchor" href="#_2-6-混合积"><span>2.6 混合积</span></a></h3><p>对于三个矢量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>，定义混合积为</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><p>混合积满足</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><h4 id="_2-6-1-混合积的几何意义" tabindex="-1"><a class="header-anchor" href="#_2-6-1-混合积的几何意义"><span>2.6.1 混合积的几何意义</span></a></h4><p>如图，以三个矢量为边的棱作平行六面体，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="ORD">∥</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-texclass="ORD">∥</mo></math></mjx-assistive-mml></mjx-container>可以视作以<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>和<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>为边的平行四边形的面积，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>可以视作这个平行六面体的体积</p><figure><img src="/assets/vector_trip-heHpDjlq.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><h3 id="_2-7-三重积" tabindex="-1"><a class="header-anchor" href="#_2-7-三重积"><span>2.7 三重积</span></a></h3><p>对于三个矢量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>，有如下公式</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:20.221em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:20.221em;" side="right"><mjx-table style="width:auto;min-width:12.953em;margin:0 3.634em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:20.221em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:1.347em;"><mjx-mtd id="mjx-eqn:2.6.1"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c36"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:1.347em;vertical-align:-0.25em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(2.6.1)</mtext></mtd><mtd><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>证明：<br> 设<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.427em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>d</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，可知<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.427em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>d</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>垂直于<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>和<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>所组成的平面的法线，也就是<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.427em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>d</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>平行于<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>和<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>所组成的平面。所以<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.427em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>d</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>可以表达成<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>和<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>的线性组合</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.427em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mover><mi>d</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mi>m</mi><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mi>n</mi><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><p>由于<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.427em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>d</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>垂直于<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>，所以</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:9.455em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mo stretchy="false">(</mo><mi>m</mi><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mi>n</mi><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mi>m</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>+</mo><mi>n</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>所以一定存在一个非零的实数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></mjx-assistive-mml></mjx-container>，使得<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>,</mo><mi>n</mi><mo>=</mo><mo>−</mo><mi>p</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，所以</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.427em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mover><mi>d</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mi>p</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><p>由于这是一个恒等式,<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></mjx-assistive-mml></mjx-container>的值和<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>无关，当特殊情况<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.272em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>时，有</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>−</mo><mi>p</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><p>两边同时点积<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>，可以得到</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>−</mo><mi>p</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><p>等号左侧应用混合积公式，可以得到</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:14.124em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mo data-mjx-texclass="ORD">∥</mo><mi>a</mi><msup><mo data-mjx-texclass="ORD">∥</mo><mn>2</mn></msup><mo data-mjx-texclass="ORD">∥</mo><mi>b</mi><msup><mo data-mjx-texclass="ORD">∥</mo><mn>2</mn></msup><mi>s</mi><mi>i</mi><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>等号右侧</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:26.357em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.265em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base style="padding-top:0.001em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.214em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2225"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mi>p</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>−</mo><mi>p</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mi>p</mi><mo data-mjx-texclass="ORD">∥</mo><mi>a</mi><msup><mo data-mjx-texclass="ORD">∥</mo><mn>2</mn></msup><mo data-mjx-texclass="ORD">∥</mo><mi>b</mi><msup><mo data-mjx-texclass="ORD">∥</mo><mn>2</mn></msup><mo>−</mo><mi>p</mi><mo data-mjx-texclass="ORD">∥</mo><mi>a</mi><msup><mo data-mjx-texclass="ORD">∥</mo><mn>2</mn></msup><mo data-mjx-texclass="ORD">∥</mo><mi>b</mi><msup><mo data-mjx-texclass="ORD">∥</mo><mn>2</mn></msup><mi>c</mi><mi>o</mi><msup><mi>s</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mi>p</mi><mo data-mjx-texclass="ORD">∥</mo><mi>a</mi><msup><mo data-mjx-texclass="ORD">∥</mo><mn>2</mn></msup><mo data-mjx-texclass="ORD">∥</mo><mi>b</mi><msup><mo data-mjx-texclass="ORD">∥</mo><mn>2</mn></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>c</mi><mi>o</mi><msup><mi>s</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mi>p</mi><mo data-mjx-texclass="ORD">∥</mo><mi>a</mi><msup><mo data-mjx-texclass="ORD">∥</mo><mn>2</mn></msup><mo data-mjx-texclass="ORD">∥</mo><mi>b</mi><msup><mo data-mjx-texclass="ORD">∥</mo><mn>2</mn></msup><mi>s</mi><mi>i</mi><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>由此可得<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container></p></div><!--[--><!--]--></div><footer class="vp-page-meta"><!----><div class="vp-meta-item git-info"><div class="vp-meta-item last-updated"><span class="meta-item-label">Last Updated: </span><!----></div><div class="vp-meta-item contributors"><span class="meta-item-label">Contributors: </span><span class="meta-item-info"><!--[--><!--[--><span class="contributor" title="email: thejinchao@gmail.com">thejinchao</span><!----><!--]--><!--]--></span></div></div></footer><nav class="vp-page-nav" aria-label="page 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