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

Mar 6, 2017

The vector is =〈-1,-7,-2〉

#### Explanation:

The vector perpendicular to 2 vectors is calculated with the determinant (cross product)

$| \left(\vec{i} , \vec{j} , \vec{k}\right) , \left(d , e , f\right) , \left(g , h , i\right) |$

where 〈d,e,f〉 and 〈g,h,i〉 are the 2 vectors

Here, we have veca=〈3,-1,2〉 and vecb=〈1,-1,3〉

Therefore,

$| \left(\vec{i} , \vec{j} , \vec{k}\right) , \left(3 , - 1 , 2\right) , \left(1 , - 1 , 3\right) |$

$= \vec{i} | \left(- 1 , 2\right) , \left(- 1 , 3\right) | - \vec{j} | \left(3 , 2\right) , \left(1 , 3\right) | + \vec{k} | \left(3 , - 1\right) , \left(1 , - 1\right) |$

$= \vec{i} \left(- 1\right) - \vec{j} \left(7\right) + \vec{k} \left(- 2\right)$

=〈-1,-7,-2〉=vecc

Verification by doing 2 dot products

$\vec{a} . \vec{c}$

=〈3,-1,2>.〈-1,-7,-2〉=-3+7-4=0

$\vec{b} . \vec{c}$

=〈1,-1,3〉.〈-1,-7,-2〉=-1+7-6=0

So,

$\vec{c}$ is perpendicular to $\vec{a}$ and $\vec{b}$