Given two vectors a, b ∈ R3: a x b (cross product). Show that the cross product is skew symmetric. If AT = −A which means A is skew symmetric then prove that (A+B) is also skew symmetric. I managed to prove it like this: (A+B)T = AT+BT =(−A+−B)=−(A+B) Therefore (A+B)T =−(A+B) which is skew symmetric.
Is this correct?
If $a=(a_{1},a_{2},a_{3})$ and $b=(b_{1},b_{2},b_{3})$ then the cross product $a\times b$ is given by: $$a\times b = \mbox{det}\begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{bmatrix} $$ Thus, if you interchange $a$ and $b$ you interchange the second and third rows of the above matrix, which culminates in a change of sign. This proves the cross product is skew-symmetric. You can explicitly calculate it, if you prefer.