Answer
$a \times b=-b \times a$
Work Step by Step
Let $a=\lt a_1,a_2,a_3\gt$ and $b=\lt b_1,b_2,b_3\gt$
$a \times b=\begin{vmatrix} i&j&k \\ a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}$
$b \times a=\begin{vmatrix} i&j&k \\b_1&b_2&b_3 \\a_1&a_2&a_3\end{vmatrix}$
Switching two rows in a determinant reverses the sign of the determinant.
$\begin{vmatrix} i&j&k \\a_1&a_2&a_3 \\b_1&b_2&b_3\end{vmatrix}=-\begin{vmatrix} i&j&k \\b_1&b_2&b_3 \\a_1&a_2&a_3\end{vmatrix}$
Hence, $a \times b=-b \times a$
