# What is the cross product of <-2, 5 ,-2 > and <7 , -9 ,2 >?

Jan 29, 2017

The answer is $= < - 8 , - 10 , - 17 >$

#### Explanation:

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

$| \left(\begin{matrix}\hat{i} & \hat{j} & \hat{k} \\ - 2 & 5 & - 2 \\ 7 & - 9 & 2\end{matrix}\right) |$

$= \hat{i} \cdot | \left(\begin{matrix}5 & - 2 \\ - 9 & 2\end{matrix}\right) | - \hat{j} \cdot | \left(\begin{matrix}- 2 & - 2 \\ 7 & 2\end{matrix}\right) | + \hat{k} \cdot | \left(\begin{matrix}- 2 & 5 \\ 7 & - 9\end{matrix}\right) |$

$= \hat{i} \left(- 8\right) - \hat{j} \left(10\right) + \hat{k} \left(- 17\right)$

$= < - 8 , - 10 , - 17 >$

Verification by doing the dot products

$< - 8 , - 10 , - 17 > . < - 2 , 5 , - 2 \ge 16 - 50 + 34 = 0$

$< - 8 , - 10 , - 17 > . < 7 , - 9 , 2 \ge - 56 + 90 - 34 = 0$