Vector product

of a vector $a$ by a vector $b$ in $\mathbb{R}^3$

The vector $c$, denoted by the symbol $a\times b$ or $[a,b]$, satisfying the following requirements:

1. the length of $c$ is equal to the product of the lengths of the vectors $a$ and $b$ by the sine of the angle $\phi$ between them, i.e. $$|c| = |a\times b| = |a|\cdot |b| \sin\phi;$$
2. $c$ is orthogonal to both $a$ and $b$;
3. the orientation of the vector triple $a,b,c$ is the same as that of the (standard) triple of basis vectors. See Vector algebra.

Let $a=(a_1,a_2,a_3)$ and $b=(b_1,b_2,b_3)$ have coordinates with respect to an orthonormal basis in $\mathbb{R}^3$, then the coordinates of $c=a\times b$ are $$c=\begin{pmatrix} a_2 b_3 - a_3 b_2 \\ a_3 b_1 - a_1 b_3 \\ a_1 b_2 - a_2 b_1\end{pmatrix}.$$