In $\Bbb R^3$, the cross product of two vectors $v$ and $w$ produces a vector $v \times w$ perpendicular to both. This tag is not meant for products in other mathematical contexts, such as products of groups (such as the [tag:direct-product]), sets (the Cartesian product), graphs, and so on.

In mathematics, the cross product, vector product, or Gibbs' vector product, is a binary operation on two vectors in $3$-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied, and therefore normal to the plane containing them.

We write:

$$\vec{u}\times\vec{v}=\begin{pmatrix}u_1\\ u_2\\ u_3\end{pmatrix}\times\begin{pmatrix}v_1 \\ v_2 \\ v_3\end{pmatrix}=\begin{pmatrix}u_2v_3-u_3v_2 \\ u_3v_1-u_1v_3 \\ u_1v_2-u_2v_1\end{pmatrix} $$

The cross product is anti-commutative, so $\vec{u}\times\vec{v}=-(\vec{v}\times\vec{u})$.

The norm of the cross-product has several important geometric properties. For instance, $||\vec{u}\times \vec{v}||$ is the area of the parallelogram spanned by $\vec{u},\vec{v}$, i.e. $||\vec{u}\times \vec{v}||=||\vec{u}||||\vec{v}||\sin(\theta)$ where $\theta$ is the angle between them. This in turn yields Lagrange's identity $$ ||\vec{u}\times \vec{v}||^2 + |\vec{u}\cdot\vec{v}|^2 = (||\vec{u}||||\vec{v}||)^2 $$If $\vec{u},\vec{v},\vec{w}\in\mathbb{R}^3$, the volume of the parallelipiped spanned by them is $|(\vec{u}\times\vec{v})\cdot \vec{w}|$.

There is also a seven-dimensional cross product as bilinear operation on vectors in seven dimensional Euclidean space.