# What is the cross product of [2, -1, 1] and [3,-6,4] ?

Apr 19, 2018

The vector is =〈2,-5,-9〉

#### 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 veca=〈d,e,f〉 and vecb=〈g,h,i〉 are the 2 vectors

Here, we have veca=〈2,-1,1〉 and vecb=〈3,-6,4〉

Therefore,

$| \left(\vec{i} , \vec{j} , \vec{k}\right) , \left(2 , - 1 , 1\right) , \left(3 , - 6 , 4\right) |$

$= \vec{i} | \left(- 1 , 1\right) , \left(- 6 , 4\right) | - \vec{j} | \left(2 , 1\right) , \left(3 , 4\right) | + \vec{k} | \left(2 , - 1\right) , \left(3 , - 6\right) |$

$= \vec{i} \left(\left(- 1\right) \cdot \left(4\right) - \left(- 6\right) \cdot \left(1\right)\right) - \vec{j} \left(\left(2\right) \cdot \left(4\right) - \left(1\right) \cdot \left(3\right)\right) + \vec{k} \left(\left(2\right) \cdot \left(- 6\right) - \left(- 1\right) \cdot \left(3\right)\right)$

=〈2,-5,-9〉=vecc

Verification by doing 2 dot products

〈2,-5,-9〉.〈2,-1,1〉=(2)*(2)+(-5)*(-1)+(-9)*(1)=0

〈2,-5,-9〉.〈3,-6,4〉=(2)*(3)+(-5)*(-6)+(-9)*(4)=0

So,

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