# What is the cross product of [1,2,1] and [3,1,-5] ?

Jan 2, 2016

$\left[1 , 2 , 1\right] \times \left[3 , 1 , - 5\right] = \left[- 11 , 8 , - 5\right]$

#### Explanation:

In general:

$\left[{a}_{x} , {a}_{y} , {a}_{z}\right] \times \left[{b}_{x} , {b}_{y} , {b}_{z}\right] = \left[\left\mid \begin{matrix}{a}_{y} & {a}_{z} \\ {b}_{y} & {b}_{z}\end{matrix} \right\mid , \left\mid \begin{matrix}{a}_{z} & {a}_{z} \\ {b}_{z} & {b}_{x}\end{matrix} \right\mid , \left\mid \begin{matrix}{a}_{x} & {a}_{y} \\ {b}_{x} & {b}_{y}\end{matrix} \right\mid\right]$

So:

$\left[1 , 2 , 1\right] \times \left[3 , 1 , - 5\right]$

$= \left[\left\mid \begin{matrix}2 & 1 \\ 1 & - 5\end{matrix} \right\mid , \left\mid \begin{matrix}1 & 1 \\ - 5 & 3\end{matrix} \right\mid , \left\mid \begin{matrix}1 & 2 \\ 3 & 1\end{matrix} \right\mid\right]$

$= \left[\left(2 \cdot - 5\right) - \left(1 \cdot 1\right) , \left(1 \cdot 3\right) - \left(1 \cdot - 5\right) , \left(1 \cdot 1\right) - \left(2 \cdot 3\right)\right]$

$= \left[- 10 - 1 , 3 + 5 , 1 - 6\right]$

$= \left[- 11 , 8 , - 5\right]$