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

Jul 31, 2017

The vector is =〈-7,-4,1〉

#### Explanation:

The cross product of 2 vectors is calculated with the determinant

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

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

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

Therefore,

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

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

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

=〈-7,-4,1〉=vecc

Verification by doing 2 dot products

〈1,-2,-1〉.〈-7,-4,1〉=-7*1+2*4-1*1=0

〈1,-2,-1〉.〈1,-1,3〉=1*1+1*2-1*3=0

So,

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