# What is the cross product of [3, 0, 5] and [1,2,1] ?

Dec 7, 2016

$\left(\begin{matrix}3 \\ 0 \\ 5\end{matrix}\right) \times \left(\begin{matrix}1 \\ 2 \\ 1\end{matrix}\right) = \left(\begin{matrix}- 10 \\ 2 \\ 6\end{matrix}\right)$, or $\left[- 10 , 2 , 6\right]$

#### Explanation:

We can use the notation:
$\setminus \setminus \setminus \setminus \setminus \left(\begin{matrix}3 \\ 0 \\ 5\end{matrix}\right) \times \left(\begin{matrix}1 \\ 2 \\ 1\end{matrix}\right) = | \left(\underline{\hat{i}} , \underline{\hat{j}} , \underline{\hat{k}}\right) , \left(3 , 0 , 5\right) , \left(1 , 2 , 1\right) |$

$\therefore \left(\begin{matrix}3 \\ 0 \\ 5\end{matrix}\right) \times \left(\begin{matrix}1 \\ 2 \\ 1\end{matrix}\right) = | \left(0 , 5\right) , \left(2 , 1\right) | \underline{\hat{i}} - | \left(3 , 5\right) , \left(1 , 1\right) | \underline{\hat{j}} + | \left(3 , 0\right) , \left(1 , 2\right) | \underline{\hat{k}}$

$\therefore \left(\begin{matrix}3 \\ 0 \\ 5\end{matrix}\right) \times \left(\begin{matrix}1 \\ 2 \\ 1\end{matrix}\right) = \left(0 - 10\right) \underline{\hat{i}} - \left(3 - 5\right) \underline{\hat{j}} + \left(6 - 0\right) \underline{\hat{k}}$

$\therefore \left(\begin{matrix}3 \\ 0 \\ 5\end{matrix}\right) \times \left(\begin{matrix}1 \\ 2 \\ 1\end{matrix}\right) = - 10 \underline{\hat{i}} + 2 \underline{\hat{j}} + 6 \underline{\hat{k}}$
$\therefore \left(\begin{matrix}3 \\ 0 \\ 5\end{matrix}\right) \times \left(\begin{matrix}1 \\ 2 \\ 1\end{matrix}\right) = \left(\begin{matrix}- 10 \\ 2 \\ 6\end{matrix}\right)$