向量叉乘算子、点乘算子与矩阵运算的关系

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−a3a2a30−a1−a2a10][b1b2b3]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×]aa×b=-[b×]aa×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 =c1c2c3
>> test2=A*B*C
test2 =a2*b1*c2 - c1*(a2*b2 + a3*b3) + a3*b1*c3a1*b2*c1 - c2*(a1*b1 + a3*b3) + a3*b2*c3a1*b3*c1 - c3*(a1*b1 + a2*b2) + a2*b3*c2
>> simplify(test1-test2)
ans =000>> 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*c1a2*b1*c1 - a1*b1*c2 + a2*b3*c3 - a3*b3*c2a3*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 =a1a2a3
>> B=[b1;b2;b3]
B =b1b2b3
>> 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 =000

结论

a×(b×c)=[0−a3a2a30−a1−a2a10][0−b3b2b30−b1−b2b10][c1c2c3]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

算子

测试

>> I=[Ixx Ixy Ixz;Ixy Iyy Iyz;Ixz Iyz Izz]
I =
[ Ixx, Ixy, Ixz]
[ Ixy, Iyy, Iyz]
[ Ixz, Iyz, Izz]>> I_=[Ixx Ixy Ixz Iyy Iyz Izz]'
I_ =IxxIxyIxzIyyIyzIzz>> w_
w_ =
[ c1, c2, c3,  0,  0,  0]
[  0, c1,  0, c2, c3,  0]
[  0,  0, c1,  0, c2, c3]>> C_ = [0 -c3 c2;
c3 0 -c1;
-c2 c1 0]
C_ =
[   0, -c3,  c2]
[  c3,   0, -c1]
[ -c2,  c1,   0]>> simplify(C_*w_*I_-cross(C,I*C))
ans =000

结论