# What is the cross product of <5, 2 ,5 > and <4 ,1 ,9 >?

Jul 4, 2016

$< 5 , 2 , 5 > \times < 4 , 1 , 9 > = < 13 , - 25 , - 3 >$

#### Explanation:

The cross product of two $3$ dimensional vectors can be defined by the formula:

$< {u}_{1} , {u}_{2} , {u}_{3} > \times < {v}_{1} , {v}_{2} , {v}_{3} > = < \left\mid \begin{matrix}{u}_{2} & {u}_{3} \\ {v}_{2} & {v}_{3}\end{matrix} \right\mid , \left\mid \begin{matrix}{u}_{3} & {u}_{1} \\ {v}_{3} & {v}_{1}\end{matrix} \right\mid , \left\mid \begin{matrix}{u}_{1} & {u}_{2} \\ {v}_{1} & {v}_{2}\end{matrix} \right\mid >$

In our example we find:

$< 5 , 2 , 5 > \times < 4 , 1 , 9 >$

$= < \left\mid \begin{matrix}2 & 5 \\ 1 & 9\end{matrix} \right\mid , \left\mid \begin{matrix}5 & 5 \\ 9 & 4\end{matrix} \right\mid , \left\mid \begin{matrix}5 & 2 \\ 4 & 1\end{matrix} \right\mid >$

$= < \left(18 - 5\right) , \left(20 - 45\right) , \left(5 - 8\right) >$

$= < 13 , - 25 , - 3 >$