cross product

The cross product of $\vec{u}$ and $\vec{v}$ is notated $\vec{u}\times \vec{v}$ and results in a vector perpendicular to both vectors. The direction of the vector can be determined by the right hand rule.

Cross product is not commutative

$\vec{A}\times \vec{B}=-\vec{B}\times \vec{A}$

Calculate $\vec{u}\times \vec{v}$ with determinant

Find the determinant of a matrix formed by vertically concatenating $⟨\hat{i},\hat{j},\hat{k}⟩$, $\vec{u}$, and $\vec{v}$:

$$\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ u_x& u_y& u_z\\ v_x& v_y& v_z\end{array}\right|=(u_y{v}_z-v_y{u}_z)\cdot \hat{i}-\dots$$

Calculate $|\vec{u}\times \vec{v}|$ with trigonometry

$|\vec{u}\times \vec{v}|=|\vec{u}||\vec{v}|\sin \theta$ Use the right hand rule to determine direction.

This is useful if knowing the actual vector is not needed or if components of the operands are unknown.