# What is the cross product of [2, 5, 4] and [4,3,6] ?

Feb 18, 2016

$< 2 , 5 , 4 > \times < 4 , 3 , 6 > = < 18 , 4 , - 14 >$

#### Explanation:

The cross product of $< {a}_{x} , {a}_{y} , {a}_{z} > \times < {b}_{x} , {b}_{y} , {b}_{z} >$ can be evaluated as:
$\left\{\begin{matrix}{c}_{x} = {a}_{y} {b}_{z} - {b}_{y} {a}_{z} \\ {c}_{y} = {a}_{z} {b}_{x} - {b}_{z} {a}_{x} \\ {c}_{z} = {a}_{x} {b}_{y} - {b}_{x} {a}_{y}\end{matrix}\right.$
$\textcolor{w h i t e}{\text{XXX}}$if you have trouble remembering the order of these combinations see below

Given
$\left.\begin{matrix}{a}_{x} & {a}_{y} & {a}_{z} \\ 2 & 5 & 4\end{matrix}\right.$ and $\left.\begin{matrix}{b}_{x} & {b}_{y} & {b}_{z} \\ 4 & 3 & 6\end{matrix}\right.$

${c}_{x} = 5 \times 6 - 3 \times 4 = 30 - 12 = 18$
${c}_{y} = 4 \times 4 - 6 \times 2 = 16 - 12 = 4$
${c}_{z} = 2 \times 3 - 4 \times 5 = 6 - 20 = - 14$

This is the "below" mentioned above (skip if not needed)
One way to remember the order of the cross product combinations is to treat the system as if it we like calculating a determinant for
something like:
$\textcolor{w h i t e}{\text{XXX}} | \left({c}_{x} , {c}_{y} , {c}_{z}\right) , \left(, = ,\right) , \left({a}_{x} , {a}_{y} , {a}_{z}\right) , \left({b}_{x} , {b}_{y} , {b}_{z}\right) |$

to get something like:
$\textcolor{w h i t e}{\text{XXX}} {c}_{x} = + | \left({a}_{y} , {a}_{z}\right) , \left({b}_{x} , {b}_{z}\right) |$

$\textcolor{w h i t e}{\text{XXX}} {c}_{y} = - | \left({a}_{x} , {a}_{z}\right) , \left({b}_{x} , {b}_{z}\right) |$

$\textcolor{w h i t e}{\text{XXX}} {c}_{z} = + | \left({a}_{x} , {a}_{y}\right) , \left({b}_{x} , {b}_{y}\right) |$
Don't forget to alternate the signs and remember this is just a memory aid not real determination evaluation!