向量叉乘算子、点乘算子与矩阵运算的关系 |
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向量叉乘算子、点乘算子与矩阵运算的关系
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文章目录
向量叉乘运算测试结论
矩阵与向量点乘变量算子测试结论
向量叉乘
( a × b ) × c = b ( a ⋅ c ) − a ( b ⋅ c ) (a×b)×c = b(a·c) - a(b·c) (a×b)×c=b(a⋅c)−a(b⋅c) a × ( b × c ) = b ( a ⋅ c ) − c ( a ⋅ b ) a×(b×c) = b(a·c) - c(a·b) a×(b×c)=b(a⋅c)−c(a⋅b) a × b = [ 0 − a 3 a 2 a 3 0 − a 1 − a 2 a 1 0 ] [ b 1 b 2 b 3 ] 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×b=⎣⎡0a3−a2−a30a1a2−a10⎦⎤⎣⎡b1b2b3⎦⎤ a × b = − [ b × ] a a×b=-[b×]a a×b=−[b×]a 运算测试 >> syms x1 y1 z1 x2 y2 z2 x3 y3 z3 real >> clear >> syms a1 a2 a3 b1 b2 b3 x y z real >> clear >> syms a1 a2 a3 b1 b2 b3 c1 c2 c3 real >> A_=[a1;a2;a3]; >> B_ = [b1; b2; b3]; >> C_ = [c1;c2;c3]; >> test1 = cross(A_,cross(B_,C_)) test1 = a2*(b1*c2 - b2*c1) + a3*(b1*c3 - b3*c1) a3*(b2*c3 - b3*c2) - a1*(b1*c2 - b2*c1) - a1*(b1*c3 - b3*c1) - a2*(b2*c3 - b3*c2) >> A = [0 -a3 a2;a3 0 -a1;-a2 a1 0] A = [ 0, -a3, a2] [ a3, 0, -a1] [ -a2, a1, 0] >> B = [0 -b3 b2;b3 0 -b1;-b2 b1 0] B = [ 0, -b3, b2] [ b3, 0, -b1] [ -b2, b1, 0] >> C=C_ C = c1 c2 c3 >> test2=A*B*C test2 = a2*b1*c2 - c1*(a2*b2 + a3*b3) + a3*b1*c3 a1*b2*c1 - c2*(a1*b1 + a3*b3) + a3*b2*c3 a1*b3*c1 - c3*(a1*b1 + a2*b2) + a2*b3*c2 >> simplify(test1-test2) ans = 0 0 0 >> test3=cross(cross(A_,B_),C_) test3 = - c2*(a1*b2 - a2*b1) - c3*(a1*b3 - a3*b1) c1*(a1*b2 - a2*b1) - c3*(a2*b3 - a3*b2) c1*(a1*b3 - a3*b1) + c2*(a2*b3 - a3*b2) >> simplify(test2-test3) ans = a1*b2*c2 - a2*b2*c1 + a1*b3*c3 - a3*b3*c1 a2*b1*c1 - a1*b1*c2 + a2*b3*c3 - a3*b3*c2 a3*b1*c1 - a1*b1*c3 - a2*b2*c3 + a3*b2*c2 ############################################################ ############ a×b=-[b×]a ############### ############################################################ >> syms a1 a2 a3 b1 b2 b3 real >> A=[a1;a2;a3] A = a1 a2 a3 >> B=[b1;b2;b3] B = b1 b2 b3 >> B_=[0 -b3 b2; b3 0 -b1; -b2 b1 0] B_ = [ 0, -b3, b2] [ b3, 0, -b1] [ -b2, b1, 0] >> simplify(cross(A,B)-(-B_*A)) ans = 0 0 0 结论a × ( b × c ) = [ 0 − a 3 a 2 a 3 0 − a 1 − a 2 a 1 0 ] [ 0 − b 3 b 2 b 3 0 − b 1 − b 2 b 1 0 ] [ c 1 c 2 c 3 ] 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] a×(b×c)=⎣⎡0a3−a2−a30a1a2−a10⎦⎤⎣⎡0b3−b2−b30b1b2−b10⎦⎤⎣⎡c1c2c3⎦⎤ 矩阵与向量点乘 变量
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