Cross product

A way to multiply two vectors that returns a new vector perpendicular to both.

The cross product $\mathbf{u}\times\mathbf{v}$ answers "what direction is perpendicular to both of these?" and, unlike the dot product, hands back another vector rather than a number. Order matters: $\mathbf{v}\times\mathbf{u} = -(\mathbf{u}\times\mathbf{v})$ (the right-hand rule fixes the direction), and $\mathbf{u}\times\mathbf{v}=\mathbf{0}$ exactly when $\mathbf{u}$ and $\mathbf{v}$ are parallel. The magnitude $|\mathbf{u}\times\mathbf{v}|$ equals the area of the parallelogram the two vectors span.

Examples

For $\mathbf{u}=\langle 2,1,-1\rangle$ , $\mathbf{v}=\langle -1,3,2\rangle$ : $\mathbf{u}\times\mathbf{v}=\langle 5,-3,7\rangle$ , spanning a parallelogram of area $\sqrt{83}$ .
Torque, $\boldsymbol{\tau}=\mathbf{r}\times\mathbf{F}$ , is a cross product.