# What are cross products?

Mar 15, 2018

See explanation...

#### Explanation:

When you encounter vectors in $3$ dimensions then you meet two ways of multiplying two vectors together:

Dot product

Written $\vec{u} \cdot \vec{v}$, this takes two vectors and produces a scalar result.

If $\vec{u} = < {u}_{1} , {u}_{2} , {u}_{3} >$ and $\vec{v} = < {v}_{1} , {v}_{2} , {v}_{3} >$ then:

$\vec{u} \cdot \vec{v} = {u}_{1} {v}_{1} + {u}_{2} {v}_{2} + {u}_{3} {v}_{3}$

Cross product

Written $\vec{u} \times \vec{v}$, this takes two vectors and produces a vector perpendicular to both of them, or the zero vector if $\vec{u}$ and $\vec{v}$ are parallel.

If $\vec{u} = < {u}_{1} , {u}_{2} , {u}_{3} >$ and $\vec{v} = < {v}_{1} , {v}_{2} , {v}_{3} >$ then:

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

This is sometimes described in terms of a determinant of a $3 \times 3$ matrix and the three unit vectors $\hat{i}$, $\hat{j}$, $\hat{k}$:

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

How about division ?

Neither dot product nor cross product allow division of vectors. To find how to divide vectors you can look at the quaternions. The quaternions form a $4$ dimensional vector space over the real numbers and have arithmetic with non-commutative multiplication that can be expressed as a combination of dot product and cross product. Actually that's the wrong way around, since quaternion arithmetic predates the modern presentation of vectors, dot and cross products.

Anyway, we can say that a quaternion can be written as a combination of a scalar part and vector part, with arithmetic defined by:

$\left({r}_{1} , \vec{{v}_{1}}\right) + \left({r}_{2} , \vec{{v}_{2}}\right) = \left({r}_{1} + {r}_{2} , \vec{{v}_{1}} + \vec{{v}_{2}}\right)$

$\left({r}_{1} , \vec{{v}_{1}}\right) \cdot \left({r}_{2} , \vec{{v}_{2}}\right) = \left({r}_{1} {r}_{2} - \vec{{v}_{1}} \cdot \vec{{v}_{2}} , {r}_{1} \vec{{v}_{2}} + {r}_{2} \vec{{v}_{1}} + \vec{{v}_{1}} \times \vec{{v}_{2}}\right)$

For a very interesting related talk, watch this...