# What is the cross product of [2, 1, -4] and [3, 2, 5] ?

Dec 19, 2015

$\left(13 , - 22 , 1\right)$

#### Explanation:

By definition, the vector cross product of these two 3-dimensional vectors in ${\mathbb{R}}^{3}$ may be given by the following matrix determinant :

$\left(2 , 1 , - 4\right) \times \left(3 , 2 , 5\right) = | \left(\hat{i} , \hat{j} , \hat{k}\right) , \left(2 , 1 , - 4\right) , \left(3 , 2 , 5\right) |$

$= \hat{i} \left(5 + 8\right) - \hat{j} \left(10 + 12\right) + \hat{k} \left(4 - 3\right)$

$= 13 \hat{i} - 22 \hat{j} + \hat{k}$

$= \left(13 , - 22 , 1\right)$