# What is the cross product of [1, 3, 4] and [3,2, 5] ?

Feb 17, 2016

$< 7 , 7 , - 7 >$

#### Explanation:

There are a couple ways to do this.
Here is one:

The cross product of $< {a}_{x} , {a}_{y} , {a}_{z} > \times < {b}_{x} , {b}_{y} , {b}_{z} > = < {c}_{x} , {c}_{y} , {c}_{z} >$
where$\left\{\begin{matrix}{c}_{x} = {a}_{y} {b}_{z} - {a}_{z} {b}_{y} \\ {c}_{y} = {a}_{z} {b}_{x} - {a}_{x} {b}_{y} \\ {c}_{z} = {a}_{x} {b}_{y} - {a}_{y} {b}_{x}\end{matrix}\right.$

Using this method:
with $\left.\begin{matrix}{a}_{x} & {a}_{y} & {a}_{z} & \null & {b}_{x} & {b}_{y} & {b}_{z} \\ 1 & 3 & 4 & \null & 3 & 2 & 5\end{matrix}\right.$

${c}_{x} = 3 \times 5 - 4 \times 2 = 7$
${c}_{b} = 4 \times 3 - 1 \times 5 = 7$
${c}_{z} = 1 \times 2 - 3 \times 3 = - 7$