import{_ as d,c as T,a as t,b as Q,d as l,o as m}from"./app-CqutohVF.js";const n="/assets/axis_left-D307nuvU.svg",e="/assets/axis_right-DlLDxrsw.svg",o="/assets/vector_op-EGHzijwY.svg",s="/assets/standard_basis-DLyK0dsc.svg",r="/assets/dot_product-LEv427NC.svg",g="/assets/cross_product-2IEbK7Oi.svg",h="/assets/vector_trip-heHpDjlq.svg",p={},i={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},H={style:{"vertical-align":"-3.733ex"},xmlns:"http://www.w3.org/2000/svg",width:"26.591ex",height:"8.597ex",role:"img",focusable:"false",viewBox:"0 -2150 11753.3 3800","aria-hidden":"true"},c={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},L={style:{"vertical-align":"-3.733ex"},xmlns:"http://www.w3.org/2000/svg",width:"24.179ex",height:"8.597ex",role:"img",focusable:"false",viewBox:"0 -2150 10687 3800","aria-hidden":"true"},x={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},u={style:{"vertical-align":"-3.733ex"},xmlns:"http://www.w3.org/2000/svg",width:"19.489ex",height:"8.597ex",role:"img",focusable:"false",viewBox:"0 -2150 8614.3 3800","aria-hidden":"true"},f={class:"MathJax",jax:"SVG",style:{position:"relative"}},M={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"26.964ex",height:"1.984ex",role:"img",focusable:"false",viewBox:"0 -683 11917.9 877","aria-hidden":"true"},Z={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},D={style:{"vertical-align":"-3.733ex"},xmlns:"http://www.w3.org/2000/svg",width:"19.489ex",height:"8.597ex",role:"img",focusable:"false",viewBox:"0 -2150 8614.3 3800","aria-hidden":"true"},w={class:"MathJax",jax:"SVG",style:{position:"relative"}},V={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"26.964ex",height:"1.984ex",role:"img",focusable:"false",viewBox:"0 -683 11917.9 877","aria-hidden":"true"},v={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},j={style:{"vertical-align":"-4.14ex"},xmlns:"http://www.w3.org/2000/svg",width:"35.649ex",height:"9.412ex",role:"img",focusable:"false",viewBox:"0 -2330 15757 4160","aria-hidden":"true"},y={class:"MathJax",jax:"SVG",style:{position:"relative"}},b={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"38.345ex",height:"3.048ex",role:"img",focusable:"false",viewBox:"0 -1097 16948.4 1347","aria-hidden":"true"},O={class:"MathJax",jax:"SVG",style:{position:"relative"}},R={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.294ex",height:"1.937ex",role:"img",focusable:"false",viewBox:"0 -845 572 856","aria-hidden":"true"},A={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},X={style:{"vertical-align":"-1.359ex"},xmlns:"http://www.w3.org/2000/svg",width:"21.373ex",height:"4.208ex",role:"img",focusable:"false",viewBox:"0 -1259.2 9447 1860","aria-hidden":"true"},C={class:"MathJax",jax:"SVG",style:{position:"relative"}},k={style:{"vertical-align":"-0.564ex"},xmlns:"http://www.w3.org/2000/svg",width:"2.552ex",height:"2.476ex",role:"img",focusable:"false",viewBox:"0 -845 1128 1094.5","aria-hidden":"true"},E={class:"MathJax",jax:"SVG",style:{position:"relative"}},B={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.294ex",height:"1.937ex",role:"img",focusable:"false",viewBox:"0 -845 572 856","aria-hidden":"true"},F={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},S={style:{"vertical-align":"-2.387ex"},xmlns:"http://www.w3.org/2000/svg",width:"29.309ex",height:"5.828ex",role:"img",focusable:"false",viewBox:"0 -1521 12954.4 2576","aria-hidden":"true"},G={class:"MathJax",jax:"SVG",style:{position:"relative"}},J={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"3.397ex",height:"2.353ex",role:"img",focusable:"false",viewBox:"0 -846 1501.7 1040","aria-hidden":"true"},N={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},z={style:{"vertical-align":"-0.667ex"},xmlns:"http://www.w3.org/2000/svg",width:"25.761ex",height:"2.581ex",role:"img",focusable:"false",viewBox:"0 -846 11386.4 1141","aria-hidden":"true"},P={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},I={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"18.354ex",height:"3.048ex",role:"img",focusable:"false",viewBox:"0 -1097 8112.3 1347","aria-hidden":"true"},q={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},Y={style:{"vertical-align":"-0.887ex"},xmlns:"http://www.w3.org/2000/svg",width:"24.815ex",height:"2.887ex",role:"img",focusable:"false",viewBox:"0 -883.9 10968 1275.9","aria-hidden":"true"},K={class:"MathJax",jax:"SVG",style:{position:"relative"}},U={style:{"vertical-align":"-0.023ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.197ex",height:"1.934ex",role:"img",focusable:"false",viewBox:"0 -845 529 855","aria-hidden":"true"},W={class:"MathJax",jax:"SVG",style:{position:"relative"}},$={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"0.971ex",height:"2.507ex",role:"img",focusable:"false",viewBox:"0 -1097 429 1108","aria-hidden":"true"},_={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},Q1={style:{"vertical-align":"-2.172ex"},xmlns:"http://www.w3.org/2000/svg",width:"9.885ex",height:"6.183ex",role:"img",focusable:"false",viewBox:"0 -1773 4369.3 2733","aria-hidden":"true"},a1={class:"MathJax",jax:"SVG",style:{position:"relative"}},T1={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"3.83ex",height:"2.376ex",role:"img",focusable:"false",viewBox:"0 -856 1692.7 1050","aria-hidden":"true"},t1={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},m1={style:{"vertical-align":"-2.412ex"},xmlns:"http://www.w3.org/2000/svg",width:"17.209ex",height:"5.878ex",role:"img",focusable:"false",viewBox:"0 -1532 7606.6 2598","aria-hidden":"true"},l1={class:"MathJax",jax:"SVG",style:{position:"relative"}},d1={style:{"vertical-align":"-0.09ex"},xmlns:"http://www.w3.org/2000/svg",width:"8.174ex",height:"2.005ex",role:"img",focusable:"false",viewBox:"0 -846 3613 886","aria-hidden":"true"},n1={class:"MathJax",jax:"SVG",style:{position:"relative"}},e1={style:{"vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"8.174ex",height:"2.1ex",role:"img",focusable:"false",viewBox:"0 -846 3613 928","aria-hidden":"true"},o1={class:"MathJax",jax:"SVG",style:{position:"relative"}},s1={style:{"vertical-align":"-0.09ex"},xmlns:"http://www.w3.org/2000/svg",width:"8.174ex",height:"2.005ex",role:"img",focusable:"false",viewBox:"0 -846 3613 886","aria-hidden":"true"},r1={class:"MathJax",jax:"SVG",style:{position:"relative"}},g1={style:{"vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"10.621ex",height:"2.667ex",role:"img",focusable:"false",viewBox:"0 -1097 4694.4 1179","aria-hidden":"true"},h1={class:"MathJax",jax:"SVG",style:{position:"relative"}},p1={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"22.703ex",height:"3.048ex",role:"img",focusable:"false",viewBox:"0 -1097 10034.8 1347","aria-hidden":"true"},i1={class:"MathJax",jax:"SVG",style:{position:"relative"}},H1={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"26.257ex",height:"3.048ex",role:"img",focusable:"false",viewBox:"0 -1097 11605.4 1347","aria-hidden":"true"},c1={class:"MathJax",jax:"SVG",style:{position:"relative"}},L1={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"3.173ex",height:"2.921ex",role:"img",focusable:"false",viewBox:"0 -1097 1402.7 1291","aria-hidden":"true"},x1={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},u1={style:{"vertical-align":"-6.141ex"},xmlns:"http://www.w3.org/2000/svg",width:"55.881ex",height:"13.414ex",role:"img",focusable:"false",viewBox:"0 -3214.4 24699.2 5928.9","aria-hidden":"true"},f1={class:"MathJax",jax:"SVG",style:{position:"relative"}},M1={style:{"vertical-align":"-0.018ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.613ex",height:"1.952ex",role:"img",focusable:"false",viewBox:"0 -855 713 863","aria-hidden":"true"},Z1={class:"MathJax",jax:"SVG",style:{position:"relative"}},D1={style:{"vertical-align":"-0.018ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.541ex",height:"1.952ex",role:"img",focusable:"false",viewBox:"0 -855 681 863","aria-hidden":"true"},w1={class:"MathJax",jax:"SVG",style:{position:"relative"}},V1={style:{"vertical-align":"-0.018ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.283ex",height:"1.955ex",role:"img",focusable:"false",viewBox:"0 -856 567 864","aria-hidden":"true"},v1={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},j1={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"23.544ex",height:"2.502ex",role:"img",focusable:"false",viewBox:"0 -856 10406.7 1106","aria-hidden":"true"},y1={class:"MathJax",jax:"SVG",style:{position:"relative"}},b1={style:{"vertical-align":"-0.023ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.197ex",height:"1.934ex",role:"img",focusable:"false",viewBox:"0 -845 529 855","aria-hidden":"true"},O1={class:"MathJax",jax:"SVG",style:{position:"relative"}},R1={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"0.971ex",height:"2.507ex",role:"img",focusable:"false",viewBox:"0 -1097 429 1108","aria-hidden":"true"},A1={class:"MathJax",jax:"SVG",style:{position:"relative"}},X1={style:{"vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"14.644ex",height:"2.667ex",role:"img",focusable:"false",viewBox:"0 -1097 6472.4 1179","aria-hidden":"true"},C1={class:"MathJax",jax:"SVG",style:{position:"relative"}},k1={style:{"vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"9.308ex",height:"2.604ex",role:"img",focusable:"false",viewBox:"0 -1069 4114 1151","aria-hidden":"true"},E1={class:"MathJax",jax:"SVG",style:{position:"relative"}},B1={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"26.097ex",height:"3.048ex",role:"img",focusable:"false",viewBox:"0 -1097 11534.8 1347","aria-hidden":"true"},F1={class:"MathJax",jax:"SVG",style:{position:"relative"}},S1={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"18.761ex",height:"3.048ex",role:"img",focusable:"false",viewBox:"0 -1097 8292.4 1347","aria-hidden":"true"},G1={class:"MathJax",jax:"SVG",style:{position:"relative"}},J1={style:{"vertical-align":"-0.023ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.197ex",height:"1.934ex",role:"img",focusable:"false",viewBox:"0 -845 529 855","aria-hidden":"true"},N1={class:"MathJax",jax:"SVG",style:{position:"relative"}},z1={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"0.971ex",height:"2.507ex",role:"img",focusable:"false",viewBox:"0 -1097 429 1108","aria-hidden":"true"},P1={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},I1={style:{"vertical-align":"-4.961ex"},xmlns:"http://www.w3.org/2000/svg",width:"63.955ex",height:"11.054ex",role:"img",focusable:"false",viewBox:"0 -2693 28268.2 4886","aria-hidden":"true"},q1={class:"MathJax",jax:"SVG",style:{position:"relative"}},Y1={style:{"vertical-align":"-0.023ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.197ex",height:"1.934ex",role:"img",focusable:"false",viewBox:"0 -845 529 855","aria-hidden":"true"},K1={class:"MathJax",jax:"SVG",style:{position:"relative"}},U1={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.122ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 1822.1 1000","aria-hidden":"true"},W1={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},$1={style:{"vertical-align":"-3.835ex"},xmlns:"http://www.w3.org/2000/svg",width:"27.257ex",height:"8.801ex",role:"img",focusable:"false",viewBox:"0 -2195 12047.6 3889.9","aria-hidden":"true"},_1={class:"MathJax",jax:"SVG",style:{position:"relative"}},Q3={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.159ex",height:"2.921ex",role:"img",focusable:"false",viewBox:"0 -1097 2280.3 1291","aria-hidden":"true"},a3={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},T3={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"9.307ex",height:"3.048ex",role:"img",focusable:"false",viewBox:"0 -1097 4113.9 1347","aria-hidden":"true"},t3={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},m3={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"33.957ex",height:"3.048ex",role:"img",focusable:"false",viewBox:"0 -1097 15008.8 1347","aria-hidden":"true"},l3={class:"MathJax",jax:"SVG",style:{position:"relative"}},d3={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"7.196ex",height:"3.048ex",role:"img",focusable:"false",viewBox:"0 -1097 3180.4 1347","aria-hidden":"true"},n3={class:"MathJax",jax:"SVG",style:{position:"relative"}},e3={style:{"vertical-align":"-0.023ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.197ex",height:"1.934ex",role:"img",focusable:"false",viewBox:"0 -845 529 855","aria-hidden":"true"},o3={class:"MathJax",jax:"SVG",style:{position:"relative"}},s3={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"0.971ex",height:"2.507ex",role:"img",focusable:"false",viewBox:"0 -1097 429 1108","aria-hidden":"true"},r3={class:"MathJax",jax:"SVG",style:{position:"relative"}},g3={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"9.307ex",height:"3.048ex",role:"img",focusable:"false",viewBox:"0 -1097 4113.9 1347","aria-hidden":"true"},h3={class:"MathJax",jax:"SVG",style:{position:"relative"}},p3={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.159ex",height:"2.921ex",role:"img",focusable:"false",viewBox:"0 -1097 2280.3 1291","aria-hidden":"true"},i3={class:"MathJax",jax:"SVG",display:"true",width:"full",style:{"min-width":"45.748ex",position:"relative"}},H3={style:{"vertical-align":"-0.958ex","min-width":"45.748ex"},xmlns:"http://www.w3.org/2000/svg",width:"100%",height:"3.048ex",role:"img",focusable:"false","aria-hidden":"true"},c3={class:"MathJax",jax:"SVG",style:{position:"relative"}},L3={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"14.632ex",height:"3.048ex",role:"img",focusable:"false",viewBox:"0 -1097 6467.4 1347","aria-hidden":"true"},x3={class:"MathJax",jax:"SVG",style:{position:"relative"}},u3={style:{"vertical-align":"-0.023ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.176ex",height:"2.505ex",role:"img",focusable:"false",viewBox:"0 -1097 520 1107","aria-hidden":"true"},f3={class:"MathJax",jax:"SVG",style:{position:"relative"}},M3={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"0.971ex",height:"2.507ex",role:"img",focusable:"false",viewBox:"0 -1097 429 1108","aria-hidden":"true"},Z3={class:"MathJax",jax:"SVG",style:{position:"relative"}},D3={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"0.98ex",height:"1.937ex",role:"img",focusable:"false",viewBox:"0 -845 433 856","aria-hidden":"true"},w3={class:"MathJax",jax:"SVG",style:{position:"relative"}},V3={style:{"vertical-align":"-0.023ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.176ex",height:"2.505ex",role:"img",focusable:"false",viewBox:"0 -1097 520 1107","aria-hidden":"true"},v3={class:"MathJax",jax:"SVG",style:{position:"relative"}},j3={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"0.971ex",height:"2.507ex",role:"img",focusable:"false",viewBox:"0 -1097 429 1108","aria-hidden":"true"},y3={class:"MathJax",jax:"SVG",style:{position:"relative"}},b3={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"0.98ex",height:"1.937ex",role:"img",focusable:"false",viewBox:"0 -845 433 856","aria-hidden":"true"},O3={class:"MathJax",jax:"SVG",style:{position:"relative"}},R3={style:{"vertical-align":"-0.023ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.176ex",height:"2.505ex",role:"img",focusable:"false",viewBox:"0 -1097 520 1107","aria-hidden":"true"},A3={class:"MathJax",jax:"SVG",style:{position:"relative"}},X3={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"0.971ex",height:"2.507ex",role:"img",focusable:"false",viewBox:"0 -1097 429 1108","aria-hidden":"true"},C3={class:"MathJax",jax:"SVG",style:{position:"relative"}},k3={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"0.98ex",height:"1.937ex",role:"img",focusable:"false",viewBox:"0 -845 433 856","aria-hidden":"true"},E3={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},B3={style:{"vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"12.253ex",height:"2.667ex",role:"img",focusable:"false",viewBox:"0 -1097 5416 1179","aria-hidden":"true"},F3={class:"MathJax",jax:"SVG",style:{position:"relative"}},S3={style:{"vertical-align":"-0.023ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.176ex",height:"2.505ex",role:"img",focusable:"false",viewBox:"0 -1097 520 1107","aria-hidden":"true"},G3={class:"MathJax",jax:"SVG",style:{position:"relative"}},J3={style:{"vertical-align":"-0.023ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.197ex",height:"1.934ex",role:"img",focusable:"false",viewBox:"0 -845 529 855","aria-hidden":"true"},N3={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},z3={style:{"vertical-align":"-2.821ex"},xmlns:"http://www.w3.org/2000/svg",width:"21.391ex",height:"6.774ex",role:"img",focusable:"false",viewBox:"0 -1747 9454.9 2994","aria-hidden":"true"},P3={class:"MathJax",jax:"SVG",style:{position:"relative"}},I3={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.138ex",height:"1.439ex",role:"img",focusable:"false",viewBox:"0 -442 503 636","aria-hidden":"true"},q3={class:"MathJax",jax:"SVG",style:{position:"relative"}},Y3={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"25.554ex",height:"3.048ex",role:"img",focusable:"false",viewBox:"0 -1097 11294.7 1347","aria-hidden":"true"},K3={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},U3={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"35.775ex",height:"3.048ex",role:"img",focusable:"false",viewBox:"0 -1097 15812.3 1347","aria-hidden":"true"},W3={class:"MathJax",jax:"SVG",style:{position:"relative"}},$3={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.138ex",height:"1.439ex",role:"img",focusable:"false",viewBox:"0 -442 503 636","aria-hidden":"true"},_3={class:"MathJax",jax:"SVG",style:{position:"relative"}},Q4={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.159ex",height:"2.921ex",role:"img",focusable:"false",viewBox:"0 -1097 2280.3 1291","aria-hidden":"true"},a4={class:"MathJax",jax:"SVG",style:{position:"relative"}},T4={style:{"vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.194ex",height:"2.097ex",role:"img",focusable:"false",viewBox:"0 -845 2295.6 927","aria-hidden":"true"},t4={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},m4={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"32.233ex",height:"3.048ex",role:"img",focusable:"false",viewBox:"0 -1097 14246.8 1347","aria-hidden":"true"},l4={class:"MathJax",jax:"SVG",style:{position:"relative"}},d4={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"0.971ex",height:"2.507ex",role:"img",focusable:"false",viewBox:"0 -1097 429 1108","aria-hidden":"true"},n4={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},e4={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"43.568ex",height:"3.048ex",role:"img",focusable:"false",viewBox:"0 -1097 19257.1 1347","aria-hidden":"true"},o4={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},s4={style:{"vertical-align":"-2.58ex"},xmlns:"http://www.w3.org/2000/svg",width:"31.954ex",height:"6.292ex",role:"img",focusable:"false",viewBox:"0 -1640.5 14123.5 2780.9","aria-hidden":"true"},r4={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},g4={style:{"vertical-align":"-4.202ex"},xmlns:"http://www.w3.org/2000/svg",width:"59.631ex",height:"9.536ex",role:"img",focusable:"false",viewBox:"0 -2357.4 26357 4214.9","aria-hidden":"true"},h4={class:"MathJax",jax:"SVG",style:{position:"relative"}},p4={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.286ex",height:"1.946ex",role:"img",focusable:"false",viewBox:"0 -666 2336.6 860","aria-hidden":"true"};function i4(H4,a){return m(),T("div",null,[a[237]||(a[237]=t('

数学基础(一)


1. 基础约定

1.1 坐标系

1.1.1 左手坐标系

DirectX一般缺省使用左手坐标系

1.1.2 右手坐标系

OpenGL缺省使用右手坐标系,图形学中缺省使用这种坐标系

1.2 左乘和右乘

1.2.1 左乘,前乘(pre-multiply)

矢量为行向量,和矩阵的乘法为

',13)),Q("mjx-container",i,[(m(),T("svg",H,a[0]||(a[0]=[t('',1)]))),a[1]||(a[1]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mo",{stretchy:"false"},"["),Q("mi",null,"x"),Q("mo",null,","),Q("mi",null,"y"),Q("mo",null,","),Q("mi",null,"z"),Q("mo",{stretchy:"false"},"]"),Q("mrow",{"data-mjx-texclass":"INNER"},[Q("mo",{"data-mjx-texclass":"OPEN"},"["),Q("mtable",{columnspacing:"1em",rowspacing:"4pt"},[Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"11")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"12")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"13")])])])]),Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"21")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"22")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"23")])])])]),Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"31")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"32")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"33")])])])])]),Q("mo",{"data-mjx-texclass":"CLOSE"},"]")])])],-1))]),a[238]||(a[238]=Q("p",null,"DirectX缺省使用左乘",-1)),a[239]||(a[239]=Q("h4",{id:"_1-2-2-右乘-后乘-post-multiply",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_1-2-2-右乘-后乘-post-multiply"},[Q("span",null,"1.2.2 右乘,后乘(post-multiply)")])],-1)),a[240]||(a[240]=Q("p",null,"矢量为列向量,和矩阵的乘法为",-1)),Q("mjx-container",c,[(m(),T("svg",L,a[2]||(a[2]=[t('',1)]))),a[3]||(a[3]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mrow",{"data-mjx-texclass":"INNER"},[Q("mo",{"data-mjx-texclass":"OPEN"},"["),Q("mtable",{columnspacing:"1em",rowspacing:"4pt"},[Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"11")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"12")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"13")])])])]),Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"21")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"22")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"23")])])])]),Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"31")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"32")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"33")])])])])]),Q("mo",{"data-mjx-texclass":"CLOSE"},"]")]),Q("mrow",{"data-mjx-texclass":"INNER"},[Q("mo",{"data-mjx-texclass":"OPEN"},"["),Q("mtable",{columnspacing:"1em",rowspacing:"4pt"},[Q("mtr",null,[Q("mtd",null,[Q("mi",null,"x")])]),Q("mtr",null,[Q("mtd",null,[Q("mi",null,"y")])]),Q("mtr",null,[Q("mtd",null,[Q("mi",null,"z")])])]),Q("mo",{"data-mjx-texclass":"CLOSE"},"]")])])],-1))]),a[241]||(a[241]=Q("p",null,"OpenGL缺省使用右乘,图形学中缺省使用这种方式。",-1)),a[242]||(a[242]=Q("h3",{id:"_1-3-行存储和列存储",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_1-3-行存储和列存储"},[Q("span",null,"1.3 行存储和列存储")])],-1)),a[243]||(a[243]=Q("h4",{id:"_1-3-1-行存储",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_1-3-1-行存储"},[Q("span",null,"1.3.1 行存储")])],-1)),a[244]||(a[244]=Q("p",null,"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",-1)),Q("mjx-container",x,[(m(),T("svg",u,a[4]||(a[4]=[t('',1)]))),a[5]||(a[5]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mrow",{"data-mjx-texclass":"INNER"},[Q("mo",{"data-mjx-texclass":"OPEN"},"["),Q("mtable",{columnspacing:"1em",rowspacing:"4pt"},[Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"11")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"12")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"13")])])])]),Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"21")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"22")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"23")])])])]),Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"31")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"32")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"33")])])])])]),Q("mo",{"data-mjx-texclass":"CLOSE"},"]")])])],-1))]),Q("p",null,[a[8]||(a[8]=l("在内存中为")),Q("mjx-container",f,[(m(),T("svg",M,a[6]||(a[6]=[t('',1)]))),a[7]||(a[7]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"11")])]),Q("mo",null,","),Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"12")])]),Q("mo",null,","),Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"13")])]),Q("mo",null,","),Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"21")])]),Q("mo",null,","),Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"22")])]),Q("mo",null,"…")])],-1))])]),a[245]||(a[245]=Q("h4",{id:"_1-3-2-列存储",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_1-3-2-列存储"},[Q("span",null,"1.3.2 列存储")])],-1)),a[246]||(a[246]=Q("p",null,"按照列优先存储一个矩阵,OpenGL使用这种方式 比如对于矩阵",-1)),Q("mjx-container",Z,[(m(),T("svg",D,a[9]||(a[9]=[t('',1)]))),a[10]||(a[10]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mrow",{"data-mjx-texclass":"INNER"},[Q("mo",{"data-mjx-texclass":"OPEN"},"["),Q("mtable",{columnspacing:"1em",rowspacing:"4pt"},[Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"11")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"12")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"13")])])])]),Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"21")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"22")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"23")])])])]),Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"31")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"32")])])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"33")])])])])]),Q("mo",{"data-mjx-texclass":"CLOSE"},"]")])])],-1))]),Q("p",null,[a[13]||(a[13]=l("在内存中为")),Q("mjx-container",w,[(m(),T("svg",V,a[11]||(a[11]=[t('',1)]))),a[12]||(a[12]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"11")])]),Q("mo",null,","),Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"21")])]),Q("mo",null,","),Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"31")])]),Q("mo",null,","),Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"12")])]),Q("mo",null,","),Q("msub",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"22")])]),Q("mo",null,"…")])],-1))])]),a[247]||(a[247]=t('

2. 矢量

2.1 基本操作

2.1.1 定义

',3)),Q("mjx-container",v,[(m(),T("svg",j,a[14]||(a[14]=[t('',1)]))),a[15]||(a[15]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mtable",{displaystyle:"true",columnalign:"right left",columnspacing:"0em",rowspacing:"3pt"},[Q("mtr",null,[Q("mtd",null,[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])])]),Q("mtd",null,[Q("mi"),Q("mo",null,"="),Q("mo",{stretchy:"false"},"["),Q("msub",null,[Q("mi",null,"u"),Q("mi",null,"x")]),Q("mo",null,","),Q("msub",null,[Q("mi",null,"u"),Q("mi",null,"y")]),Q("mo",null,","),Q("msub",null,[Q("mi",null,"u"),Q("mi",null,"z")]),Q("msup",null,[Q("mo",{stretchy:"false"},"]"),Q("mi",null,"T")]),Q("mo",null,","),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"v"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mo",{stretchy:"false"},"["),Q("msub",null,[Q("mi",null,"v"),Q("mi",null,"x")]),Q("mo",null,","),Q("msub",null,[Q("mi",null,"v"),Q("mi",null,"y")]),Q("mo",null,","),Q("msub",null,[Q("mi",null,"v"),Q("mi",null,"z")]),Q("msup",null,[Q("mo",{stretchy:"false"},"]"),Q("mi",null,"T")])])]),Q("mtr",null,[Q("mtd",null,[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"±"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"v"),Q("mo",{stretchy:"false"},"→")])])]),Q("mtd",null,[Q("mi"),Q("mo",null,"="),Q("mo",{stretchy:"false"},"["),Q("msub",null,[Q("mi",null,"u"),Q("mi",null,"x")]),Q("mo",null,"±"),Q("msub",null,[Q("mi",null,"v"),Q("mi",null,"x")]),Q("mo",null,","),Q("msub",null,[Q("mi",null,"u"),Q("mi",null,"y")]),Q("mo",null,"±"),Q("msub",null,[Q("mi",null,"v"),Q("mi",null,"y")]),Q("mo",null,","),Q("msub",null,[Q("mi",null,"u"),Q("mi",null,"z")]),Q("mo",null,"±"),Q("msub",null,[Q("mi",null,"v"),Q("mi",null,"z")]),Q("msup",null,[Q("mo",{stretchy:"false"},"]"),Q("mi",null,"T")])])]),Q("mtr",null,[Q("mtd",null,[Q("mi",null,"k"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])])]),Q("mtd",null,[Q("mi"),Q("mo",null,"="),Q("mo",{stretchy:"false"},"["),Q("mi",null,"k"),Q("msub",null,[Q("mi",null,"u"),Q("mi",null,"x")]),Q("mo",null,","),Q("mi",null,"k"),Q("msub",null,[Q("mi",null,"u"),Q("mi",null,"y")]),Q("mo",null,","),Q("mi",null,"k"),Q("msub",null,[Q("mi",null,"u"),Q("mi",null,"z")]),Q("msup",null,[Q("mo",{stretchy:"false"},"]"),Q("mi",null,"T")])])])])])],-1))]),a[248]||(a[248]=t('

2.1.2 几何意义

2.2 标准矢量

',4)),Q("p",null,[a[18]||(a[18]=l("标准矢量(Standard Basis),定义")),Q("mjx-container",y,[(m(),T("svg",b,a[16]||(a[16]=[t('',1)]))),a[17]||(a[17]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"i"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mo",{stretchy:"false"},"["),Q("mn",null,"1"),Q("mo",null,","),Q("mn",null,"0"),Q("mo",null,","),Q("mn",null,"0"),Q("msup",null,[Q("mo",{stretchy:"false"},"]"),Q("mi",null,"T")]),Q("mo",null,","),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"j"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mo",{stretchy:"false"},"["),Q("mn",null,"0"),Q("mo",null,","),Q("mn",null,"1"),Q("mo",null,","),Q("mn",null,"0"),Q("msup",null,[Q("mo",{stretchy:"false"},"]"),Q("mi",null,"T")]),Q("mo",null,","),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"k"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mo",{stretchy:"false"},"["),Q("mn",null,"0"),Q("mo",null,","),Q("mn",null,"0"),Q("mo",null,","),Q("mn",null,"1"),Q("msup",null,[Q("mo",{stretchy:"false"},"]"),Q("mi",null,"T")])])],-1))])]),a[249]||(a[249]=Q("h3",{id:"_2-3-长度",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_2-3-长度"},[Q("span",null,"2.3 长度")])],-1)),a[250]||(a[250]=Q("h4",{id:"_2-3-1-定义",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_2-3-1-定义"},[Q("span",null,"2.3.1 定义")])],-1)),Q("p",null,[a[21]||(a[21]=l("对于三维矢量")),Q("mjx-container",O,[(m(),T("svg",R,a[19]||(a[19]=[t('',1)]))),a[20]||(a[20]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[22]||(a[22]=l(",定义矢量的长度"))]),Q("mjx-container",A,[(m(),T("svg",X,a[23]||(a[23]=[t('',1)]))),a[24]||(a[24]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mo",null,"="),Q("msqrt",null,[Q("msubsup",null,[Q("mi",null,"u"),Q("mi",null,"x"),Q("mn",null,"2")]),Q("mo",null,"+"),Q("msubsup",null,[Q("mi",null,"u"),Q("mi",null,"y"),Q("mn",null,"2")]),Q("mo",null,"+"),Q("msubsup",null,[Q("mi",null,"u"),Q("mi",null,"z"),Q("mn",null,"2")])])])],-1))]),Q("p",null,[a[27]||(a[27]=l("有时候矢量的长度也可以用")),Q("mjx-container",C,[(m(),T("svg",k,a[25]||(a[25]=[t('',1)]))),a[26]||(a[26]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|")])],-1))]),a[28]||(a[28]=l("表示,但容易和绝对值混淆"))]),a[251]||(a[251]=Q("h4",{id:"_2-3-2-定义",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_2-3-2-定义"},[Q("span",null,"2.3.2 定义")])],-1)),Q("p",null,[a[31]||(a[31]=l("对于三维矢量")),Q("mjx-container",E,[(m(),T("svg",B,a[29]||(a[29]=[t('',1)]))),a[30]||(a[30]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[32]||(a[32]=l(",定义矢量的单位矢量"))]),Q("mjx-container",F,[(m(),T("svg",S,a[33]||(a[33]=[t('',1)]))),a[34]||(a[34]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},"^")])]),Q("mo",null,"="),Q("mfrac",null,[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mrow",null,[Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{"data-mjx-texclass":"ORD"},"∥")])]),Q("mo",null,"="),Q("mo",{stretchy:"false"},"("),Q("mfrac",null,[Q("msub",null,[Q("mi",null,"u"),Q("mi",null,"x")]),Q("mrow",null,[Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{"data-mjx-texclass":"ORD"},"∥")])]),Q("mo",null,","),Q("mfrac",null,[Q("msub",null,[Q("mi",null,"u"),Q("mi",null,"y")]),Q("mrow",null,[Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{"data-mjx-texclass":"ORD"},"∥")])]),Q("mo",null,","),Q("mfrac",null,[Q("msub",null,[Q("mi",null,"u"),Q("mi",null,"z")]),Q("mrow",null,[Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{"data-mjx-texclass":"ORD"},"∥")])]),Q("mo",{stretchy:"false"},")")])],-1))]),a[252]||(a[252]=Q("h3",{id:"_2-4-点积",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_2-4-点积"},[Q("span",null,"2.4 点积")])],-1)),a[253]||(a[253]=Q("h4",{id:"_2-4-1-定义",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_2-4-1-定义"},[Q("span",null,"2.4.1 定义")])],-1)),Q("p",null,[a[37]||(a[37]=l("对于两个三维矢量")),Q("mjx-container",G,[(m(),T("svg",J,a[35]||(a[35]=[t('',1)]))),a[36]||(a[36]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,","),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"v"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[38]||(a[38]=l(",定义它们的点积(")),a[39]||(a[39]=Q("a",{href:"https://en.wikipedia.org/wiki/Dot_product",target:"_blank",rel:"noopener noreferrer"},"Dot Product",-1)),a[40]||(a[40]=l(")为"))]),Q("mjx-container",N,[(m(),T("svg",z,a[41]||(a[41]=[t('',1)]))),a[42]||(a[42]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"v"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("msub",null,[Q("mi",null,"u"),Q("mi",null,"x")]),Q("msub",null,[Q("mi",null,"v"),Q("mi",null,"x")]),Q("mo",null,"+"),Q("msub",null,[Q("mi",null,"u"),Q("mi",null,"y")]),Q("msub",null,[Q("mi",null,"v"),Q("mi",null,"y")]),Q("mo",null,"+"),Q("msub",null,[Q("mi",null,"u"),Q("mi",null,"z")]),Q("msub",null,[Q("mi",null,"v"),Q("mi",null,"z")])])],-1))]),a[254]||(a[254]=Q("h4",{id:"_2-4-2-点积的几何意义",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_2-4-2-点积的几何意义"},[Q("span",null,"2.4.2 点积的几何意义")])],-1)),a[255]||(a[255]=Q("figure",null,[Q("img",{src:r,alt:"",tabindex:"0",loading:"lazy"}),Q("figcaption")],-1)),Q("mjx-container",P,[(m(),T("svg",I,a[43]||(a[43]=[t('',1)]))),a[44]||(a[44]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mi",null,"cos"),Q("mo",{"data-mjx-texclass":"NONE"},"⁡"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"θ")])])],-1))]),a[256]||(a[256]=Q("p",null,"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",-1)),Q("mjx-container",q,[(m(),T("svg",Y,a[45]||(a[45]=[t('',1)]))),a[46]||(a[46]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mo",null,"="),Q("msubsup",null,[Q("mi",null,"u"),Q("mi",null,"x"),Q("mn",null,"2")]),Q("mo",null,"+"),Q("msubsup",null,[Q("mi",null,"u"),Q("mi",null,"y"),Q("mn",null,"2")])])],-1))]),Q("p",null,[a[51]||(a[51]=l("向量")),Q("mjx-container",K,[(m(),T("svg",U,a[47]||(a[47]=[t('',1)]))),a[48]||(a[48]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[52]||(a[52]=l("在向量")),Q("mjx-container",W,[(m(),T("svg",$,a[49]||(a[49]=[t('',1)]))),a[50]||(a[50]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[53]||(a[53]=l("上的投影的长度为"))]),Q("mjx-container",_,[(m(),T("svg",Q1,a[54]||(a[54]=[t('',1)]))),a[55]||(a[55]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"b")]),Q("mo",null,"="),Q("mfrac",null,[Q("mrow",null,[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])])]),Q("mrow",null,[Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mi",null,"b"),Q("mo",{"data-mjx-texclass":"ORD"},"∥")])])])],-1))]),a[257]||(a[257]=Q("h4",{id:"_2-4-3-推论",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_2-4-3-推论"},[Q("span",null,"2.4.3 推论")])],-1)),Q("p",null,[a[58]||(a[58]=l("判断两个矢量")),Q("mjx-container",a1,[(m(),T("svg",T1,a[56]||(a[56]=[t('',1)]))),a[57]||(a[57]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,","),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"v"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[59]||(a[59]=l("之间的夹角和它们之间点积的关系"))]),Q("mjx-container",t1,[(m(),T("svg",m1,a[60]||(a[60]=[t('',1)]))),a[61]||(a[61]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mi",null,"cos"),Q("mo",{"data-mjx-texclass":"NONE"},"⁡"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"θ"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mfrac",null,[Q("mrow",null,[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"v"),Q("mo",{stretchy:"false"},"→")])])]),Q("mrow",null,[Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"v"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{"data-mjx-texclass":"ORD"},"∥")])])])],-1))]),Q("ul",null,[Q("li",null,[a[64]||(a[64]=l("当")),Q("mjx-container",l1,[(m(),T("svg",d1,a[62]||(a[62]=[t('',1)]))),a[63]||(a[63]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"v"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,">"),Q("mn",null,"0")])],-1))]),a[65]||(a[65]=l(",夹角在[0,90)之间"))]),Q("li",null,[a[68]||(a[68]=l("当")),Q("mjx-container",n1,[(m(),T("svg",e1,a[66]||(a[66]=[t('',1)]))),a[67]||(a[67]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"v"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mn",null,"0")])],-1))]),a[69]||(a[69]=l(",两个矢量垂直"))]),Q("li",null,[a[72]||(a[72]=l("当")),Q("mjx-container",o1,[(m(),T("svg",s1,a[70]||(a[70]=[t('',1)]))),a[71]||(a[71]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"v"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"<"),Q("mn",null,"0")])],-1))]),a[73]||(a[73]=l(",两个矢量夹角在(90,180]之间"))])]),a[258]||(a[258]=Q("h4",{id:"_2-4-3-基本运算规则",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_2-4-3-基本运算规则"},[Q("span",null,"2.4.3 基本运算规则")])],-1)),a[259]||(a[259]=Q("p",null,"矢量的点积满足一般的交换律、分配律",-1)),Q("ul",null,[Q("li",null,[Q("mjx-container",r1,[(m(),T("svg",g1,a[74]||(a[74]=[t('',1)]))),a[75]||(a[75]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])])])],-1))])]),Q("li",null,[Q("mjx-container",h1,[(m(),T("svg",p1,a[76]||(a[76]=[t('',1)]))),a[77]||(a[77]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"+"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"+"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])])])],-1))])]),Q("li",null,[Q("mjx-container",i1,[(m(),T("svg",H1,a[78]||(a[78]=[t('',1)]))),a[79]||(a[79]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mo",{stretchy:"false"},"("),Q("mi",null,"k"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"k"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mi",null,"k"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")")])],-1))])])]),a[260]||(a[260]=Q("h3",{id:"_2-5叉积",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_2-5叉积"},[Q("span",null,"2.5叉积")])],-1)),a[261]||(a[261]=Q("h4",{id:"_2-5-1-定义",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_2-5-1-定义"},[Q("span",null,"2.5.1 定义")])],-1)),Q("p",null,[a[82]||(a[82]=l("对于两个三维矢量")),Q("mjx-container",c1,[(m(),T("svg",L1,a[80]||(a[80]=[t('',1)]))),a[81]||(a[81]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,","),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[83]||(a[83]=l(",定义它们的叉积"))]),Q("mjx-container",x1,[(m(),T("svg",u1,a[84]||(a[84]=[t('',1)]))),a[85]||(a[85]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mtable",{displaystyle:"true",columnalign:"right left",columnspacing:"0em",rowspacing:"3pt"},[Q("mtr",null,[Q("mtd",null,[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"b"),Q("mo",{stretchy:"false"},"→")])])]),Q("mtd",null,[Q("mi"),Q("mo",null,"="),Q("mrow",{"data-mjx-texclass":"INNER"},[Q("mo",{"data-mjx-texclass":"OPEN"},"|"),Q("mtable",{columnalign:"center",columnspacing:"1em",rowspacing:"4pt"},[Q("mtr",null,[Q("mtd",null,[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"i"),Q("mo",{stretchy:"false"},"→")])])]),Q("mtd",null,[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"j"),Q("mo",{stretchy:"false"},"→")])])]),Q("mtd",null,[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"k"),Q("mo",{stretchy:"false"},"→")])])])]),Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"x")])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"y")])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"z")])])]),Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"b"),Q("mi",null,"x")])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"b"),Q("mi",null,"y")])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"b"),Q("mi",null,"z")])])])]),Q("mo",{"data-mjx-texclass":"CLOSE"},"|")])])]),Q("mtr",null,[Q("mtd"),Q("mtd",null,[Q("mi"),Q("mo",null,"="),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"y")]),Q("msub",null,[Q("mi",null,"b"),Q("mi",null,"z")]),Q("mo",null,"−"),Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"z")]),Q("msub",null,[Q("mi",null,"b"),Q("mi",null,"y")]),Q("mo",{stretchy:"false"},")"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"i"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"+"),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"z")]),Q("msub",null,[Q("mi",null,"b"),Q("mi",null,"x")]),Q("mo",null,"−"),Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"x")]),Q("msub",null,[Q("mi",null,"b"),Q("mi",null,"z")]),Q("mo",{stretchy:"false"},")"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"j"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"+"),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"x")]),Q("msub",null,[Q("mi",null,"b"),Q("mi",null,"y")]),Q("mo",null,"−"),Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"y")]),Q("msub",null,[Q("mi",null,"b"),Q("mi",null,"x")]),Q("mo",{stretchy:"false"},")"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"k"),Q("mo",{stretchy:"false"},"→")])])])])])])],-1))]),a[262]||(a[262]=Q("h4",{id:"_2-5-2-叉积的几何意义",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_2-5-2-叉积的几何意义"},[Q("span",null,"2.5.2 叉积的几何意义")])],-1)),a[263]||(a[263]=Q("p",null,"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。",-1)),a[264]||(a[264]=Q("figure",null,[Q("img",{src:g,alt:"",tabindex:"0",loading:"lazy"}),Q("figcaption")],-1)),Q("p",null,[a[92]||(a[92]=l("定义向量")),Q("mjx-container",f1,[(m(),T("svg",M1,a[86]||(a[86]=[t('',1)]))),a[87]||(a[87]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"n"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[93]||(a[93]=l("是一个单位向量,方向同时垂直于向量")),Q("mjx-container",Z1,[(m(),T("svg",D1,a[88]||(a[88]=[t('',1)]))),a[89]||(a[89]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"u"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[94]||(a[94]=l("和")),Q("mjx-container",w1,[(m(),T("svg",V1,a[90]||(a[90]=[t('',1)]))),a[91]||(a[91]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"v"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[95]||(a[95]=l(",且符合右手定则(右手坐标系),那么"))]),Q("mjx-container",v1,[(m(),T("svg",j1,a[96]||(a[96]=[t('',1)]))),a[97]||(a[97]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"v"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"v"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mi",null,"sin"),Q("mo",{"data-mjx-texclass":"NONE"},"⁡"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"θ"),Q("mo",{stretchy:"false"},")"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",{mathvariant:"bold-italic"},"n"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),Q("p",null,[a[102]||(a[102]=l("向量a和向量b的叉积还可以视作以")),Q("mjx-container",y1,[(m(),T("svg",b1,a[98]||(a[98]=[t('',1)]))),a[99]||(a[99]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[103]||(a[103]=l("和")),Q("mjx-container",O1,[(m(),T("svg",R1,a[100]||(a[100]=[t('',1)]))),a[101]||(a[101]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[104]||(a[104]=l("为边的平行四边形的面积"))]),a[265]||(a[265]=Q("h4",{id:"_2-5-3-基本运算规则",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_2-5-3-基本运算规则"},[Q("span",null,"2.5.3 基本运算规则")])],-1)),Q("ul",null,[Q("li",null,[Q("mjx-container",A1,[(m(),T("svg",X1,a[105]||(a[105]=[t('',1)]))),a[106]||(a[106]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mo",null,"−"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])])])],-1))])]),Q("li",null,[Q("mjx-container",C1,[(m(),T("svg",k1,a[107]||(a[107]=[t('',1)]))),a[108]||(a[108]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mn",null,"0"),Q("mo",{stretchy:"false"},"→")])])])],-1))])]),Q("li",null,[Q("mjx-container",E1,[(m(),T("svg",B1,a[109]||(a[109]=[t('',1)]))),a[110]||(a[110]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"+"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"+"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])])])],-1))])]),Q("li",null,[Q("mjx-container",F1,[(m(),T("svg",S1,a[111]||(a[111]=[t('',1)]))),a[112]||(a[112]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"k"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mi",null,"k"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")")])],-1))])])]),a[266]||(a[266]=Q("h4",{id:"_2-5-4-叉积的矩阵形式",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_2-5-4-叉积的矩阵形式"},[Q("span",null,"2.5.4 叉积的矩阵形式")])],-1)),Q("p",null,[a[117]||(a[117]=l("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")),Q("mjx-container",G1,[(m(),T("svg",J1,a[113]||(a[113]=[t('',1)]))),a[114]||(a[114]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[118]||(a[118]=l("和")),Q("mjx-container",N1,[(m(),T("svg",z1,a[115]||(a[115]=[t('',1)]))),a[116]||(a[116]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[119]||(a[119]=l("的叉积"))]),Q("mjx-container",P1,[(m(),T("svg",I1,a[120]||(a[120]=[t('',1)]))),a[121]||(a[121]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mrow",{"data-mjx-texclass":"INNER"},[Q("mo",{"data-mjx-texclass":"OPEN"},"["),Q("mtable",{columnspacing:"1em",rowspacing:"4pt"},[Q("mtr",null,[Q("mtd",null,[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])])]),Q("msub",null,[Q("mo",{stretchy:"false"},")"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"x")])])])]),Q("mtr",null,[Q("mtd",null,[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])])]),Q("msub",null,[Q("mo",{stretchy:"false"},")"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"y")])])])]),Q("mtr",null,[Q("mtd",null,[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])])]),Q("msub",null,[Q("mo",{stretchy:"false"},")"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"z")])])])])]),Q("mo",{"data-mjx-texclass":"CLOSE"},"]")]),Q("mo",null,"="),Q("mrow",{"data-mjx-texclass":"INNER"},[Q("mo",{"data-mjx-texclass":"OPEN"},"["),Q("mtable",{columnspacing:"1em",rowspacing:"4pt"},[Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"y")]),Q("msub",null,[Q("mi",null,"b"),Q("mi",null,"z")]),Q("mo",null,"−"),Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"z")]),Q("msub",null,[Q("mi",null,"b"),Q("mi",null,"y")])])]),Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"z")]),Q("msub",null,[Q("mi",null,"b"),Q("mi",null,"x")]),Q("mo",null,"−"),Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"x")]),Q("msub",null,[Q("mi",null,"b"),Q("mi",null,"z")])])]),Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"x")]),Q("msub",null,[Q("mi",null,"b"),Q("mi",null,"y")]),Q("mo",null,"−"),Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"y")]),Q("msub",null,[Q("mi",null,"b"),Q("mi",null,"x")])])])]),Q("mo",{"data-mjx-texclass":"CLOSE"},"]")]),Q("mo",null,"="),Q("mrow",{"data-mjx-texclass":"INNER"},[Q("mo",{"data-mjx-texclass":"OPEN"},"["),Q("mtable",{columnspacing:"1em",rowspacing:"4pt"},[Q("mtr",null,[Q("mtd",null,[Q("mn",null,"0")]),Q("mtd",null,[Q("mo",null,"−"),Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"z")])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"y")])])]),Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"z")])]),Q("mtd",null,[Q("mn",null,"0")]),Q("mtd",null,[Q("mo",null,"−"),Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"x")])])]),Q("mtr",null,[Q("mtd",null,[Q("mo",null,"−"),Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"y")])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"x")])]),Q("mtd",null,[Q("mn",null,"0")])])]),Q("mo",{"data-mjx-texclass":"CLOSE"},"]")]),Q("mrow",{"data-mjx-texclass":"INNER"},[Q("mo",{"data-mjx-texclass":"OPEN"},"["),Q("mtable",{columnspacing:"1em",rowspacing:"4pt"},[Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"b"),Q("mi",null,"x")])])]),Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"b"),Q("mi",null,"y")])])]),Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"b"),Q("mi",null,"z")])])])]),Q("mo",{"data-mjx-texclass":"CLOSE"},"]")])])],-1))]),Q("p",null,[a[126]||(a[126]=l("这里称这个矩阵为矢量")),Q("mjx-container",q1,[(m(),T("svg",Y1,a[122]||(a[122]=[t('',1)]))),a[123]||(a[123]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[127]||(a[127]=l("的叉乘矩阵")),a[128]||(a[128]=Q("a",{href:"https://en.wikipedia.org/wiki/Cross_product#Conversion_to_matrix_multiplication",target:"_blank",rel:"noopener noreferrer"},"cross-product matrix",-1)),a[129]||(a[129]=l(",写作")),Q("mjx-container",K1,[(m(),T("svg",U1,a[124]||(a[124]=[t('',1)]))),a[125]||(a[125]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mo",{stretchy:"false"},"["),Q("mi",{mathvariant:"bold-italic"},"a"),Q("msub",null,[Q("mo",{stretchy:"false"},"]"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mo",null,"×")])])])],-1))])]),Q("mjx-container",W1,[(m(),T("svg",$1,a[130]||(a[130]=[t('',1)]))),a[131]||(a[131]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mo",{stretchy:"false"},"["),Q("mi",{mathvariant:"bold-italic"},"a"),Q("msub",null,[Q("mo",{stretchy:"false"},"]"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mo",null,"×")])]),Q("mover",null,[Q("mo",null,"="),Q("mrow",null,[Q("mi",null,"d"),Q("mi",null,"e"),Q("mi",null,"f")])]),Q("mrow",{"data-mjx-texclass":"INNER"},[Q("mo",{"data-mjx-texclass":"OPEN"},"["),Q("mtable",{columnspacing:"1em",rowspacing:"4pt"},[Q("mtr",null,[Q("mtd",null,[Q("mn",null,"0")]),Q("mtd",null,[Q("mo",null,"−"),Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"z")])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"y")])])]),Q("mtr",null,[Q("mtd",null,[Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"z")])]),Q("mtd",null,[Q("mn",null,"0")]),Q("mtd",null,[Q("mo",null,"−"),Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"x")])])]),Q("mtr",null,[Q("mtd",null,[Q("mo",null,"−"),Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"y")])]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"a"),Q("mi",null,"x")])]),Q("mtd",null,[Q("mn",null,"0")])])]),Q("mo",{"data-mjx-texclass":"CLOSE"},"]")])])],-1))]),a[267]||(a[267]=Q("h3",{id:"_2-6-混合积",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_2-6-混合积"},[Q("span",null,"2.6 混合积")])],-1)),Q("p",null,[a[134]||(a[134]=l("对于三个矢量")),Q("mjx-container",_1,[(m(),T("svg",Q3,a[132]||(a[132]=[t('',1)]))),a[133]||(a[133]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,","),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,","),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[135]||(a[135]=l(",定义混合积为"))]),Q("mjx-container",a3,[(m(),T("svg",T3,a[136]||(a[136]=[t('',1)]))),a[137]||(a[137]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[268]||(a[268]=Q("p",null,"混合积满足",-1)),Q("mjx-container",t3,[(m(),T("svg",m3,a[138]||(a[138]=[t('',1)]))),a[139]||(a[139]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[269]||(a[269]=Q("h4",{id:"_2-6-1-混合积的几何意义",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_2-6-1-混合积的几何意义"},[Q("span",null,"2.6.1 混合积的几何意义")])],-1)),Q("p",null,[a[148]||(a[148]=l("如图,以三个矢量为边的棱作平行六面体,")),Q("mjx-container",l3,[(m(),T("svg",d3,a[140]||(a[140]=[t('',1)]))),a[141]||(a[141]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{"data-mjx-texclass":"ORD"},"∥")])],-1))]),a[149]||(a[149]=l("可以视作以")),Q("mjx-container",n3,[(m(),T("svg",e3,a[142]||(a[142]=[t('',1)]))),a[143]||(a[143]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[150]||(a[150]=l("和")),Q("mjx-container",o3,[(m(),T("svg",s3,a[144]||(a[144]=[t('',1)]))),a[145]||(a[145]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[151]||(a[151]=l("为边的平行四边形的面积,")),Q("mjx-container",r3,[(m(),T("svg",g3,a[146]||(a[146]=[t('',1)]))),a[147]||(a[147]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[152]||(a[152]=l("可以视作这个平行六面体的体积"))]),a[270]||(a[270]=Q("figure",null,[Q("img",{src:h,alt:"",tabindex:"0",loading:"lazy"}),Q("figcaption")],-1)),a[271]||(a[271]=Q("h3",{id:"_2-7-三重积",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_2-7-三重积"},[Q("span",null,"2.7 三重积")])],-1)),Q("p",null,[a[155]||(a[155]=l("对于三个矢量")),Q("mjx-container",h3,[(m(),T("svg",p3,a[153]||(a[153]=[t('',1)]))),a[154]||(a[154]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,","),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,","),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[156]||(a[156]=l(",有如下公式"))]),Q("mjx-container",i3,[(m(),T("svg",H3,a[157]||(a[157]=[t('',1)]))),a[158]||(a[158]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mtable",{displaystyle:"true"},[Q("mlabeledtr",null,[Q("mtd",null,[Q("mtext",null,"(2.6.1)")]),Q("mtd",null,[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"−"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])])])])])])],-1))]),Q("p",null,[a[179]||(a[179]=l("证明:")),a[180]||(a[180]=Q("br",null,null,-1)),a[181]||(a[181]=l(" 设")),Q("mjx-container",c3,[(m(),T("svg",L3,a[159]||(a[159]=[t('',1)]))),a[160]||(a[160]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"d"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")")])],-1))]),a[182]||(a[182]=l(",可知")),Q("mjx-container",x3,[(m(),T("svg",u3,a[161]||(a[161]=[t('',1)]))),a[162]||(a[162]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"d"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[183]||(a[183]=l("垂直于")),Q("mjx-container",f3,[(m(),T("svg",M3,a[163]||(a[163]=[t('',1)]))),a[164]||(a[164]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[184]||(a[184]=l("和")),Q("mjx-container",Z3,[(m(),T("svg",D3,a[165]||(a[165]=[t('',1)]))),a[166]||(a[166]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[185]||(a[185]=l("所组成的平面的法线,也就是")),Q("mjx-container",w3,[(m(),T("svg",V3,a[167]||(a[167]=[t('',1)]))),a[168]||(a[168]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"d"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[186]||(a[186]=l("平行于")),Q("mjx-container",v3,[(m(),T("svg",j3,a[169]||(a[169]=[t('',1)]))),a[170]||(a[170]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[187]||(a[187]=l("和")),Q("mjx-container",y3,[(m(),T("svg",b3,a[171]||(a[171]=[t('',1)]))),a[172]||(a[172]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[188]||(a[188]=l("所组成的平面。所以")),Q("mjx-container",O3,[(m(),T("svg",R3,a[173]||(a[173]=[t('',1)]))),a[174]||(a[174]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"d"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[189]||(a[189]=l("可以表达成")),Q("mjx-container",A3,[(m(),T("svg",X3,a[175]||(a[175]=[t('',1)]))),a[176]||(a[176]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[190]||(a[190]=l("和")),Q("mjx-container",C3,[(m(),T("svg",k3,a[177]||(a[177]=[t('',1)]))),a[178]||(a[178]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[191]||(a[191]=l("的线性组合"))]),Q("mjx-container",E3,[(m(),T("svg",B3,a[192]||(a[192]=[t('',1)]))),a[193]||(a[193]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"d"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mi",null,"m"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"+"),Q("mi",null,"n"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),Q("p",null,[a[198]||(a[198]=l("由于")),Q("mjx-container",F3,[(m(),T("svg",S3,a[194]||(a[194]=[t('',1)]))),a[195]||(a[195]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"d"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[199]||(a[199]=l("垂直于")),Q("mjx-container",G3,[(m(),T("svg",J3,a[196]||(a[196]=[t('',1)]))),a[197]||(a[197]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[200]||(a[200]=l(",所以"))]),Q("mjx-container",N3,[(m(),T("svg",z3,a[201]||(a[201]=[t('',1)]))),a[202]||(a[202]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mtable",{displaystyle:"true",columnalign:"right left",columnspacing:"0em",rowspacing:"3pt"},[Q("mtr",null,[Q("mtd",null,[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"m"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"+"),Q("mi",null,"n"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")")]),Q("mtd",null,[Q("mi"),Q("mo",null,"="),Q("mn",null,"0")])]),Q("mtr",null,[Q("mtd",null,[Q("mi",null,"m"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"+"),Q("mi",null,"n"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")")]),Q("mtd",null,[Q("mi"),Q("mo",null,"="),Q("mn",null,"0")])])])])],-1))]),Q("p",null,[a[207]||(a[207]=l("所以一定存在一个非零的实数")),Q("mjx-container",P3,[(m(),T("svg",I3,a[203]||(a[203]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D45D",d:"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"})])])],-1)]))),a[204]||(a[204]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"p")])],-1))]),a[208]||(a[208]=l(",使得")),Q("mjx-container",q3,[(m(),T("svg",Y3,a[205]||(a[205]=[t('',1)]))),a[206]||(a[206]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"m"),Q("mo",null,"="),Q("mi",null,"p"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,","),Q("mi",null,"n"),Q("mo",null,"="),Q("mo",null,"−"),Q("mi",null,"p"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")")])],-1))]),a[209]||(a[209]=l(",所以"))]),Q("mjx-container",K3,[(m(),T("svg",U3,a[210]||(a[210]=[t('',1)]))),a[211]||(a[211]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"d"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mi",null,"p"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"−"),Q("mi",null,"p"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),Q("p",null,[a[218]||(a[218]=l("由于这是一个恒等式,")),Q("mjx-container",W3,[(m(),T("svg",$3,a[212]||(a[212]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D45D",d:"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"})])])],-1)]))),a[213]||(a[213]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"p")])],-1))]),a[219]||(a[219]=l("的值和")),Q("mjx-container",_3,[(m(),T("svg",Q4,a[214]||(a[214]=[t('',1)]))),a[215]||(a[215]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,","),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,","),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[220]||(a[220]=l("无关,当特殊情况")),Q("mjx-container",a4,[(m(),T("svg",T4,a[216]||(a[216]=[t('',1)]))),a[217]||(a[217]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"c"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[221]||(a[221]=l("时,有"))]),Q("mjx-container",t4,[(m(),T("svg",m4,a[222]||(a[222]=[t('',1)]))),a[223]||(a[223]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mi",null,"p"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"−"),Q("mi",null,"p"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),Q("p",null,[a[226]||(a[226]=l("两边同时点积")),Q("mjx-container",l4,[(m(),T("svg",d4,a[224]||(a[224]=[t('',1)]))),a[225]||(a[225]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])])])],-1))]),a[227]||(a[227]=l(",可以得到"))]),Q("mjx-container",n4,[(m(),T("svg",e4,a[228]||(a[228]=[t('',1)]))),a[229]||(a[229]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"="),Q("mi",null,"p"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"−"),Q("mi",null,"p"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")")])],-1))]),a[272]||(a[272]=Q("p",null,"等号左侧应用混合积公式,可以得到",-1)),Q("mjx-container",o4,[(m(),T("svg",s4,a[230]||(a[230]=[t('',1)]))),a[231]||(a[231]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mtable",{displaystyle:"true",columnalign:"right left",columnspacing:"0em",rowspacing:"3pt"},[Q("mtr",null,[Q("mtd",null,[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])])]),Q("mtd",null,[Q("mi"),Q("mo",null,"="),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"⋅"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"×"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")")])]),Q("mtr",null,[Q("mtd"),Q("mtd",null,[Q("mi"),Q("mo",null,"="),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mi",null,"a"),Q("msup",null,[Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mn",null,"2")]),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mi",null,"b"),Q("msup",null,[Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mn",null,"2")]),Q("mi",null,"s"),Q("mi",null,"i"),Q("msup",null,[Q("mi",null,"n"),Q("mn",null,"2")]),Q("mo",{stretchy:"false"},"("),Q("mi",null,"θ"),Q("mo",{stretchy:"false"},")")])])])])],-1))]),a[273]||(a[273]=Q("p",null,"等号右侧",-1)),Q("mjx-container",r4,[(m(),T("svg",g4,a[232]||(a[232]=[t('',1)]))),a[233]||(a[233]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mtable",{displaystyle:"true",columnalign:"right left",columnspacing:"0em",rowspacing:"3pt"},[Q("mtr",null,[Q("mtd",null,[Q("mi",null,"p"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"−"),Q("mi",null,"p"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"⋅"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"b"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},")")]),Q("mtd",null,[Q("mi"),Q("mo",null,"="),Q("mi",null,"p"),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mi",null,"a"),Q("msup",null,[Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mn",null,"2")]),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mi",null,"b"),Q("msup",null,[Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mn",null,"2")]),Q("mo",null,"−"),Q("mi",null,"p"),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mi",null,"a"),Q("msup",null,[Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mn",null,"2")]),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mi",null,"b"),Q("msup",null,[Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mn",null,"2")]),Q("mi",null,"c"),Q("mi",null,"o"),Q("msup",null,[Q("mi",null,"s"),Q("mn",null,"2")]),Q("mo",{stretchy:"false"},"("),Q("mi",null,"θ"),Q("mo",{stretchy:"false"},")")])]),Q("mtr",null,[Q("mtd"),Q("mtd",null,[Q("mi"),Q("mo",null,"="),Q("mi",null,"p"),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mi",null,"a"),Q("msup",null,[Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mn",null,"2")]),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mi",null,"b"),Q("msup",null,[Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mn",null,"2")]),Q("mo",{stretchy:"false"},"("),Q("mn",null,"1"),Q("mo",null,"−"),Q("mi",null,"c"),Q("mi",null,"o"),Q("msup",null,[Q("mi",null,"s"),Q("mn",null,"2")]),Q("mo",{stretchy:"false"},"("),Q("mi",null,"θ"),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},")")])]),Q("mtr",null,[Q("mtd"),Q("mtd",null,[Q("mi"),Q("mo",null,"="),Q("mi",null,"p"),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mi",null,"a"),Q("msup",null,[Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mn",null,"2")]),Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mi",null,"b"),Q("msup",null,[Q("mo",{"data-mjx-texclass":"ORD"},"∥"),Q("mn",null,"2")]),Q("mi",null,"s"),Q("mi",null,"i"),Q("msup",null,[Q("mi",null,"n"),Q("mn",null,"2")]),Q("mo",{stretchy:"false"},"("),Q("mi",null,"θ"),Q("mo",{stretchy:"false"},")")])])])])],-1))]),Q("p",null,[a[236]||(a[236]=l("由此可得")),Q("mjx-container",h4,[(m(),T("svg",p4,a[234]||(a[234]=[t('',1)]))),a[235]||(a[235]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"p"),Q("mo",null,"="),Q("mn",null,"1")])],-1))])])])}const L4=d(p,[["render",i4],["__file","math_01.html.vue"]]),x4=JSON.parse('{"path":"/note/graphics/math/math_01.html","title":"数学基础(一)","lang":"zh-CN","frontmatter":{},"headers":[{"level":2,"title":"1. 基础约定","slug":"_1-基础约定","link":"#_1-基础约定","children":[{"level":3,"title":"1.1 坐标系","slug":"_1-1-坐标系","link":"#_1-1-坐标系","children":[]},{"level":3,"title":"1.2 左乘和右乘","slug":"_1-2-左乘和右乘","link":"#_1-2-左乘和右乘","children":[]},{"level":3,"title":"1.3 行存储和列存储","slug":"_1-3-行存储和列存储","link":"#_1-3-行存储和列存储","children":[]}]},{"level":2,"title":"2. 矢量","slug":"_2-矢量","link":"#_2-矢量","children":[{"level":3,"title":"2.1 基本操作","slug":"_2-1-基本操作","link":"#_2-1-基本操作","children":[]},{"level":3,"title":"2.2 标准矢量","slug":"_2-2-标准矢量","link":"#_2-2-标准矢量","children":[]},{"level":3,"title":"2.3 长度","slug":"_2-3-长度","link":"#_2-3-长度","children":[]},{"level":3,"title":"2.4 点积","slug":"_2-4-点积","link":"#_2-4-点积","children":[]},{"level":3,"title":"2.5叉积","slug":"_2-5叉积","link":"#_2-5叉积","children":[]},{"level":3,"title":"2.6 混合积","slug":"_2-6-混合积","link":"#_2-6-混合积","children":[]},{"level":3,"title":"2.7 三重积","slug":"_2-7-三重积","link":"#_2-7-三重积","children":[]}]}],"git":{"contributors":[{"name":"thejinchao","username":"thejinchao","email":"thejinchao@gmail.com","commits":2,"url":"https://github.com/thejinchao"}]},"filePathRelative":"note/graphics/math/math_01.md"}');export{L4 as comp,x4 as data};