# What is the cross-product of two vectors?

Mar 15, 2018

A vector orthogonal to both of them...

#### Explanation:

The cross product of two vectors in $3$ dimensional space is a third vector orthogonal to both of them and of length proportional to the product of the lengths of the two vectors.

We write the cross product of $\vec{u} = < {u}_{1} , {u}_{2} , {u}_{3} >$ and $\vec{v} = < {v}_{1} , {v}_{2} , {v}_{3} >$ as:

$\vec{u} \times \vec{v} = < {u}_{2} {v}_{3} - {u}_{3} {v}_{2} , \textcolor{w h i t e}{.} {u}_{3} {v}_{1} - {u}_{1} {v}_{2} , \textcolor{w h i t e}{.} {u}_{1} {v}_{2} - {u}_{2} {v}_{1} >$

If the angle between the vectors $\vec{u}$ and $\vec{v}$ is $\theta$ then we find:

$\left\mid \left\mid \vec{u} \times \vec{v} \right\mid \right\mid = \left\mid \left\mid \vec{u} \right\mid \right\mid \cdot \left\mid \left\mid \vec{v} \right\mid \right\mid \textcolor{w h i t e}{.} \sin \theta$

Another way of writing the cross product is:

$\left({u}_{1} \hat{i} + {u}_{2} \hat{j} + {u}_{3} \hat{k}\right) \times \left({v}_{1} \hat{i} + {v}_{2} \hat{j} + {v}_{3} \hat{k}\right) = \left\mid \begin{matrix}\hat{i} & \hat{j} & \hat{k} \\ {u}_{1} & {u}_{2} & {u}_{3} \\ {v}_{1} & {v}_{2} & {v}_{3}\end{matrix} \right\mid$

Note that if $\vec{u}$ and $\vec{v}$ are parallel, then their cross product is the zero vector.