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

Jan 17, 2017

The cross product is =〈-1,2,-1〉

#### Explanation:

The cross product 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=〈-1,0,1〉 and vecb=〈0,1,2〉

Therefore,

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

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

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

=〈-1,2,-1〉=vecc

Verification by doing 2 dot products

〈-1,2,-1〉.〈-1,0,1〉=1+0-1=0

〈-1,2,-1〉.〈0,1,2〉=0+2-2=0

So,

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