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

May 4, 2018

$\left(0 , 18 , 6\right)$

#### Explanation:

The easiest way to write out the cross product is as a determinant. This can be written as
$\left(1 , - 1 , 3\right) \times \left(5 , 1 , - 3\right) = | \left(\hat{i} , \hat{j} , \hat{k}\right) , \left(1 , - 1 , 3\right) , \left(5 , 1 , - 3\right) |$

Calculating this,

$= \hat{i} \left(- 1 \cdot - 3 - 1 \cdot 3\right) - \hat{j} \left(1 \cdot - 3 - 5 \cdot 3\right) + \hat{k} \left(1 \cdot 1 - 5 \cdot - 1\right)$
$= - \hat{j} \left(- 3 - 15\right) + \hat{k} \left(1 + 5\right) = 18 \hat{j} + 6 \hat{k} = \left(0 , 18 , 6\right)$