常用的幾種向量運算法則

$a·b = b·a$

$a(b·c)≠(a·b)c$

$(a+b)·c = a·c+b·c$

$a×b = - b×a$

$(ra)×b=a×(rb)=r(a×b),其中r是標量$

$(a+b)×c = a×c+b×c$

$(a×b)×c = b(a·c) - a(b·c)$

$a×(b×c) = b(a·c) - c(a·b)$

$a×(b×c) ≠ (a×b)×c$

$(a×b)·(c×d) = a·[b×(c×d)]$

$a×(b×c)+b×(c×a)+c×(a×b)=0$

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

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

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