# What is the cross product of [3, 0, 5] and [2, -1, 1] ?

Apr 10, 2018

The vector is =〈5,7,-3〉

#### 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=〈3,0,5〉 and vecb=〈2,-1,1〉

Therefore,

$| \left(\vec{i} , \vec{j} , \vec{k}\right) , \left(3 , 0 , 5\right) , \left(2 , - 1 , 1\right) |$

$= \vec{i} | \left(0 , 5\right) , \left(- 1 , 1\right) | - \vec{j} | \left(3 , 5\right) , \left(2 , 1\right) | + \vec{k} | \left(3 , 0\right) , \left(2 , - 1\right) |$

$= \vec{i} \left(\left(0\right) \cdot \left(1\right) - \left(- 1\right) \cdot \left(5\right)\right) - \vec{j} \left(\left(3\right) \cdot \left(1\right) - \left(2\right) \cdot \left(5\right)\right) + \vec{k} \left(\left(3\right) \cdot \left(- 1\right) - \left(0\right) \cdot \left(2\right)\right)$

=〈5,7,-3〉=vecc

Verification by doing 2 dot products

〈5,7,-3〉.〈3,0,5〉=(5)*(3)+(7)*(0)+(-3)*(5)=0

〈5,7,-3〉.〈2,-1,1〉=(5)*(2)+(7)*(-1)+(-3)*(1)=0

So,

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