# 天行健 君子当自强而不息

## 向量的点积和叉积定义

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|u - v||u - v| = |u||u| + |v||v| - 2|u||v|cosα

===>

（ux - vx2 + (uy - vy)= ux2 + uy2 +vx2+vy2- 2|u||v|cosα

===>

-2uxvx - 2uyvy = -2|u||v|cosα

===>

cosα = (uxvx + uyvy) / (|u||v|)

u . v = 0时（即uxvx + uyvy = 0），向量uv垂直；当u . v > 0时，uv之间的夹角为锐角；当u . v < 0时，uv之间的夹角为钝角。

uxwx + uywy + uzwz = 0;
vxwx + vywy + vzwz = 0;

(uxvy - uyvx)wx = (uyvz - uzvy)wz
(uxvy - uyvx)wy = (uzvx - uxvz)wz

w = (wx, wy, wz) = ((uyvz - uzvy)wz / (uxvy - uyvx), (uzvx - uxvz)wz / (uxvy - uyvx), wz)
= (wz / (uxvy - uyvx) * (uyvz - uzvy, uzvx - uxvz, uxvy - uyvx))

ux(uyvz - uzvy) + uy(uzvx - uxvz) + uz(uxvy - uyvx)
= uxuyvz - uxuzvy + uyuzvx - uyuxvz + uzuxvy - uzuyvx
= (uxuyvz - uyuxvz) + (uyuzvx - uzuyvx) + (uzuxvy - uxuzvy)
= 0 + 0 + 0 = 0

vx(uyvz - uzvy) + vy(uzvx - uxvz) + vz(uxvy - uyvx)
= vxuyvz - vxuzvy + vyuzvx - vyuxvz + vzuxvy - vzuyvx
= (vxuyvz - vzuyvx) + (vyuzvx - vxuzvy) + (vzuxvy - vyuxvz)
= 0 + 0 + 0 = 0

i x j = (1, 0, 0) x (0, 1, 0) = (0 * 0 - 0 * 1, 0 * 0 - 1 * 0, 1 * 1 - 0 * 0) = (0, 0, 1) = k

j x k = (0, 1, 0) x (0, 0, 1) = (1 * 1 - 0 * 0, 0 * 0 - 0 * 1, 0 * 0 - 0 * 0) = (1, 0, 0) = i

k x i = (0, 0, 1) x (1, 0, 0) = (0 * 0 - 1 * 0, 1 * 1 - 0 * 0, 0 * 0 - 0 * 0) = (0, 1, 0) = j

v x u = (vyuz - vzuy, vzux - vxuz, vxuy - vyux) = - (u x v)

posted on 2007-04-26 18:34 lovedday 阅读(18762) 评论(6)  编辑 收藏 引用 所属分类: ■ 3D Math Basis

### 评论

#### #re: 向量的点积和叉积定义 2009-03-01 02:30 shenyan

wz / (uxvy - uyvx）这个为1，是二维向量的叉乘，这里没有证明。似乎不完备。
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#### #re: 向量的点积和叉积定义 2013-02-05 18:28 sansi

|u x v|=|u||v|sinα 怎么推导  回复  更多评论

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