# What is the cross product of <7, 2 ,5 > and <-3 ,1 ,-6 >?

Dec 24, 2016

$\left\langle7 , 2 , 5\right\rangle \times \left\langle- 3 , 1 , - 6\right\rangle = \left\langle- 17 , 27 , 13\right\rangle$

#### Explanation:

We can use the notation:
$\left(\begin{matrix}7 \\ 2 \\ 5\end{matrix}\right) \times \left(\begin{matrix}- 3 \\ 1 \\ - 6\end{matrix}\right) = | \left(\underline{\hat{i}} , \underline{\hat{j}} , \underline{\hat{k}}\right) , \left(7 , 2 , 5\right) , \left(- 3 , 1 , - 6\right) |$

$\text{ } = | \left(2 , 5\right) , \left(1 , - 6\right) | \underline{\hat{i}} - | \left(7 , 5\right) , \left(- 3 , - 6\right) | \underline{\hat{j}} + | \left(7 , 2\right) , \left(- 3 , 1\right) | \underline{\hat{k}}$

$\text{ } = \left\{\left(2\right) \left(- 6\right) - \left(1\right) \left(5\right)\right\} \underline{\hat{i}}$
$\text{ " - {(7)(-6)-(-3)(5)} ul(hat(j)) }$
$\text{ } + \left\{\left(7\right) \left(1\right) - \left(- 3\right) \left(2\right)\right\} \underline{\hat{k}}$

$\text{ } = \left(- 12 - 5\right) \underline{\hat{i}} - \left(- 42 + 15\right) \underline{\hat{j}} + \left(7 + 6\right) \underline{\hat{k}}$

$\text{ } = - 17 \underline{\hat{i}} + 27 \underline{\hat{j}} + 13 \underline{\hat{k}}$
$\text{ } = \left(\begin{matrix}- 17 \\ 27 \\ 13\end{matrix}\right)$