# What is the cross product of [5, 6, -3] and [5, 2, 9]?

Jan 29, 2017

The answer is $< 60 , - 60 , - 20 >$

#### Explanation:

The cross product of 2 vectors $\vec{a}$ and $\vec{b}$ is given by the determinant

$| \left(\begin{matrix}\hat{i} & \hat{j} & \hat{k} \\ 5 & 6 & - 3 \\ 5 & 2 & 9\end{matrix}\right) |$

$= \hat{i} \cdot | \left(\begin{matrix}6 & - 3 \\ 2 & 9\end{matrix}\right) | - \hat{j} \cdot | \left(\begin{matrix}5 & - 3 \\ 5 & 9\end{matrix}\right) | + \hat{k} \cdot | \left(\begin{matrix}5 & 6 \\ 5 & 2\end{matrix}\right) |$

$= \hat{i} \left(60\right) - \hat{j} \left(60\right) + \hat{k} \left(- 20\right)$

$= < 60 , - 60 , - 20 >$

Verification by doing the dot products

$< 60 , - 60 , - 20 > . < 5 , 6 , - 3 \ge 300 - 360 + 60 = 0$

$< 60 , - 60 , - 20 > . < 5 , 2 , 9 \ge 300 - 120 - 180 = 0$