常用的几种向量运算法则

article/2025/10/23 2:42:11

$a \cdot b = b \cdot a$

$a (b \cdot c) \neq = (a \cdot b) c$

$(a + b) \cdot c = a \cdot c + b \cdot c$

$a \times b = - b \times a$

$(r a) \times b = a \times (r b) = r (a \times b), 其中 r 是标量$

$(a + b) \times c = a \times c + b \times c$

$(a \times b) \times c = b (a \cdot c) - a (b \cdot c)$

$a \times (b \times c) = b (a \cdot c) - c (a \cdot b)$

$a \times (b \times c) \neq = (a \times b) \times c$

$(a \times b) \cdot (c \times d) = a \cdot [b \times (c \times d)]$

$a \times (b \times c) + b \times (c \times a) + c \times (a \times b) = 0$

$a×(b×c)=(a^{T}c)b-(a^{T}b)c=[a^{T}c-ca^{T}]b$

$\times b=\left[\begin{array}{ccc} 0 & -a_{3} & a_{2} \\ a_{3} & 0 & -a_{1} \\ -a_{2} & a_{1} & 0 \end{array}\right]\left[\begin{array}{l} b_{1} \\ b_{2} \\ b_{3} \end{array}\right]$

$\times(b \times c)=\left[\begin{array}{ccc} 0 & -a_{3} & a_{2} \\ a_{3} & 0 & -a_{1} \\ -a_{2} & a_{1} & 0 \end{array}\right]\left[\begin{array}{ccc} 0 & -b_{3} & b_{2} \\ b_{3} & 0 & -b_{1} \\ -b_{2} & b_{1} & 0 \end{array}\right]\left[\begin{array}{c} c_{1} \\ c_{2} \\ c_{3} \end{array}\right]$

代数规则
1. 反交换律： $a \times b = - b \times a$
2. 加法的分配律： $a \times (b + c) = a \times b + a \times c$
3. 与标量乘法兼容： $(r a) \times b = a \times (r b) = r (a \times b)$
4. 不满足结合律，但满足雅可比恒等式： $a \times (b \times c) + b \times (c \times a) + c \times (a \times b) = 0$
5. 分配律，线性性和雅可比恒等式别表明：具有向量加法和叉积的R3构成了一个李代数。
6. 两个非零向量a和b平行，当且仅当a×b=0。

在这里插入图片描述

验算：

clear,clc;
syms px py pz wx wy wz real;
c = [px;py;pz]
w = [wx;wy;wz]temp1 = c*c'*w-c'*c*diag([1 1 1])*w+cross(c,cross(w,c))
simplify(temp1)temp2 = cross(c,cross(w,cross(w,c))) + cross(w,c*c'*w)-cross(w,c'*c*diag([1 1 1])*w)
simplify(temp2)