# What is the cross product of << -3,-6,-3 >> and << -5,2,-7 >>?

Oct 1, 2017

The vector is =〈48,-6,-36〉

#### 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 〈d,e,f〉 and 〈g,h,i〉 are the 2 vectors

Here, we have veca=〈-3,-6,-3〉 and vecb=〈-5,2,-7〉

Therefore,

$| \left(\vec{i} , \vec{j} , \vec{k}\right) , \left(- 3 , - 6 , - 3\right) , \left(- 5 , 2 , - 7\right) |$

$= \vec{i} | \left(- 6 , - 3\right) , \left(2 , - 7\right) | - \vec{j} | \left(- 3 , - 3\right) , \left(- 5 , - 7\right) | + \vec{k} | \left(- 3 , - 6\right) , \left(- 5 , 2\right) |$

$= \vec{i} \left(\left(- 6 \cdot - 7\right) - \left(2 \cdot - 3\right)\right) - \vec{j} \left(\left(- 3 \cdot - 7\right) - \left(- 5 \cdot - 3\right)\right) + \vec{k} \left(\left(- 3 \cdot 2\right) - \left(- 6 \cdot - 5\right)\right)$

=〈48,-6,-36〉=vecc

Verification by doing 2 dot products (inner products)

〈48,-6,-36〉.〈-3,-6,-3〉=-48*3+6*6+36*3=0

〈48,-6,-36〉.〈-5,2,-7〉=-48*5-6*2+36*7=0

So,

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