cross product

The cross product is a measure for how perpendicular two vectors are.

This is just the opposite of the dot product which measures how parallel two vectors are:
Dot product: $a\cdot b=\Vert a\Vert \cdot \Vert b\Vert \cos (\theta )$
Cross product: $a\times b=||a||\cdot ||b||\sin (\theta )$
Perpendicular: $\vec{v}\times \vec{w}=\Vert a\Vert \cdot \Vert b\Vert$, $\vec{v}\cdot \vec{w}=0$
Parallel: $\vec{v}\times \vec{w}=0$, $\vec{v}\cdot \vec{w}=\Vert a\Vert \cdot \Vert b\Vert$

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The cross product is the same as the area of the parallelogram drawn between two vectors:

Computing the Cross-Product

The cross product of two vectors $\vec{v}=\left[\begin{array}{c}{v}_1\\ v_2\\ v_3\end{array}\right]$ and $\vec{w}=\left[\begin{array}{c}{w}_1\\ w_2\\ w_3\end{array}\right]$ is given by:

$$\vec{v}\times \vec{w}=\left[\begin{array}{c}{v}_2{w}_3-v_3{w}_2\\ v_3{w}_1-v_1{w}_3\\ v_1{w}_2-v_2{w}_1\end{array}\right]$$