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

Feb 22, 2017

The result is =〈2,-5,-9〉

#### 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=〈1,4,-2〉 and vecb=〈2,-1,1〉

Therefore,

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

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

$= \vec{i} \left(2\right) - \vec{j} \left(5\right) + \vec{k} \left(- 9\right)$

=〈2,-5,-9〉=vecc

Verification by doing 2 dot products

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

=〈1,4,-2>.〈2,-5,-9〉=2-20+18=0

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

=〈2,-1,1〉.〈2,-5,-9〉=4+5-9=0

So,

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