# What is the cross product of <4 , 5 ,-9 > and <4, 3 ,-3 >?

Mar 18, 2018

The vector is $< 12 , - 24 , - 8 >$

#### 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=〈4,5,-9〉 and vecb=〈4,3,-3〉

Therefore,

$| \left(\vec{i} , \vec{j} , \vec{k}\right) , \left(4 , 5 , - 9\right) , \left(4 , 3 , - 3\right) |$

$= \vec{i} | \left(5 , - 9\right) , \left(3 , - 3\right) | - \vec{j} | \left(4 , - 9\right) , \left(4 , - 3\right) | + \vec{k} | \left(4 , 5\right) , \left(4 , 3\right) |$

$= \vec{i} \left(\left(5\right) \cdot \left(- 3\right) - \left(9\right) \cdot \left(- 3\right)\right) - \vec{j} \left(\left(4\right) \cdot \left(- 3\right) - \left(9\right) \cdot \left(4\right)\right) + \vec{k} \left(\left(4\right) \cdot \left(3\right) - \left(4\right) \cdot \left(5\right)\right)$

=〈12,-24,-8〉=vecc

Verification by doing 2 dot products

〈12,-24,-8〉.〈4,5,-9〉=(12)*(4)+(-24)*(5)+(-8)*(-9)=0

〈12,-24,-8〉.〈4,3,-3〉=(12)*(4)+(-24)*(3)+(-8)*(-3)=0

So,

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