The vector cross-product is an operation that takes two vectors and produces another. It is non-associative and non-commutative.

A vector cross-product can be computed by taking the determinant of a matrix:

 ${\bf~a}\times{\bf~b}=\left|\begin{matrix}{\bf~i}&{\bf~j}&{\bf~k}\\a_1&a_2&a_3\\b_1&b_2&b_3\end{matrix}\right|={\bf~i}(a_2b_3-a_3b_2)-{\bf~j}(a_1b_3-a_3b_1)+{\bf~k}(a_1b_2-a_2b_1)=\left[\begin{matrix}a_2b_3-a_3b_2\\-(a_1b_3-a_3b_1)\\a_1b_2-a_2b_1\end{matrix}\right]$
where a is the vector $\left[\begin{matrix}a_1\\a_2\\a_3\end{matrix}\right]$ , b is the vector $\left[\begin{matrix}b_1\\b_2\\b_3\end{matrix}\right]$ and i, j and k are unit vectors.