# What is the cross product of <5, 2 ,15 > and <4 ,1 ,9 >?

Dec 10, 2016

$< 5 , 2 , 15 > \times < 4 , 1 , 9 > = < 3 , 15 , - 3 >$

#### Explanation:

We can use the notation:
$\setminus \setminus \setminus \setminus \setminus \left(\begin{matrix}5 \\ 2 \\ 15\end{matrix}\right) \times \left(\begin{matrix}4 \\ 1 \\ 9\end{matrix}\right) = | \left(\underline{\hat{i}} , \underline{\hat{j}} , \underline{\hat{k}}\right) , \left(5 , 2 , 15\right) , \left(4 , 1 , 9\right) |$

$\therefore \left(\begin{matrix}5 \\ 2 \\ 15\end{matrix}\right) \times \left(\begin{matrix}4 \\ 1 \\ 9\end{matrix}\right) = | \left(2 , 15\right) , \left(1 , 9\right) | \underline{\hat{i}} - | \left(5 , 15\right) , \left(4 , 9\right) | \underline{\hat{j}} + | \left(5 , 2\right) , \left(4 , 1\right) | \underline{\hat{k}}$

$\therefore \left(\begin{matrix}5 \\ 2 \\ 15\end{matrix}\right) \times \left(\begin{matrix}4 \\ 1 \\ 9\end{matrix}\right) = \left(18 - 15\right) \underline{\hat{i}} - \left(45 - 60\right) \underline{\hat{j}} + \left(5 - 8\right) \underline{\hat{k}}$

$\therefore \left(\begin{matrix}5 \\ 2 \\ 15\end{matrix}\right) \times \left(\begin{matrix}4 \\ 1 \\ 9\end{matrix}\right) = 3 \underline{\hat{i}} + 15 \underline{\hat{j}} - 3 \underline{\hat{k}}$
$\therefore \left(\begin{matrix}5 \\ 2 \\ 15\end{matrix}\right) \times \left(\begin{matrix}4 \\ 1 \\ 9\end{matrix}\right) = \left(\begin{matrix}3 \\ 15 \\ - 3\end{matrix}\right)$