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

Jun 13, 2016

It is $\left(- 5 , - 10 , - 5\right)$.

#### Explanation:

Given two vectors ${v}_{1} = \left({x}_{1} , {y}_{1} , {z}_{1}\right)$ and ${v}_{2} = \left({x}_{2} , {y}_{2} , {z}_{2}\right)$ the cross product

${v}_{1} \setminus \times {v}_{2}$ is given by

${v}_{1} \setminus \times {v}_{2} = \left({y}_{1} {z}_{2} - {z}_{1} {y}_{2} , {z}_{1} {x}_{2} - {x}_{1} {z}_{2} , {x}_{1} {y}_{2} - {x}_{2} {y}_{1}\right)$.

For your two vectors it is

$\left(4 , - 3 , 2\right) \setminus \times \left(1 , - 2 , 3\right)$
$= \left(- 3 \cdot 3 - 2 \cdot \left(- 2\right) , 2 \cdot 1 - 4 \cdot 3 , 4 \cdot \left(- 2\right) - \left(- 3\right) \cdot 1\right)$
$= \left(- 9 + 4 , 2 - 12 , - 8 + 3\right)$
$= \left(- 5 , - 10 , - 5\right)$.