# What is the cross product of (2i -3j + 4k) and (4 i + 4 j + 2 k)?

May 13, 2018

The vector is =〈-22,12,20〉

#### 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,-3,4〉 and vecb=〈4,4,2〉

Therefore,

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

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

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

=〈-22,12,20〉=vecc

Verification by doing 2 dot products

〈-22,12,20〉.〈2,-3,4〉=(-22)*(2)+(12)*(-3)+(20)*(4)=0

〈-22,12,20〉.〈4,4,2〉=(-22)*(4)+(12)*(4)+(20)*(2)=0

So,

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