# What is the cross product of [2, -1, 4] and [5, 2, -2] ?

Feb 25, 2017

$< 2 , - 1 , 4 > \times < 5 , 2 , - 2 > = < - 6 , 24 , 9 >$

#### Explanation:

We can use the notation:

$\setminus \setminus \setminus \setminus \setminus \left(\begin{matrix}2 \\ - 1 \\ 4\end{matrix}\right) \times \left(\begin{matrix}5 \\ 2 \\ - 2\end{matrix}\right) = | \left(\underline{\hat{i}} , \underline{\hat{j}} , \underline{\hat{k}}\right) , \left(2 , - 1 , 4\right) , \left(5 , 2 , - 2\right) |$

$\text{ } = | \left(- 1 , 4\right) , \left(2 , - 2\right) | \underline{\hat{i}} - | \left(2 , 4\right) , \left(5 , - 2\right) | \underline{\hat{j}} + | \left(2 , - 1\right) , \left(5 , 2\right) | \underline{\hat{k}}$

$\text{ } = \left(2 - 8\right) \underline{\hat{i}} - \left(- 4 - 20\right) \underline{\hat{j}} + \left(4 + 5\right) \underline{\hat{k}}$

$\text{ } = - 6 \underline{\hat{i}} + 24 \underline{\hat{j}} + 9 \underline{\hat{k}}$
$\text{ } = \left(\begin{matrix}- 6 \\ 24 \\ 9\end{matrix}\right)$