# What is the cross product of <7 ,4 ,-3 > and <-5 ,2 , -7 >?

Apr 20, 2018

The vector is =〈-22,64,34〉

#### 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=〈7,4,-3〉 and vecb=〈-5,2,-7〉

Therefore,

$| \left(\vec{i} , \vec{j} , \vec{k}\right) , \left(7 , 4 , - 3\right) , \left(- 5 , 2 , - 7\right) |$

$= \vec{i} | \left(4 , - 3\right) , \left(2 , - 7\right) | - \vec{j} | \left(7 , - 3\right) , \left(- 5 , - 7\right) | + \vec{k} | \left(7 , 4\right) , \left(- 5 , 2\right) |$

$= \vec{i} \left(\left(4\right) \cdot \left(- 7\right) - \left(- 3\right) \cdot \left(2\right)\right) - \vec{j} \left(\left(7\right) \cdot \left(- 7\right) - \left(- 3\right) \cdot \left(5\right)\right) + \vec{k} \left(\left(7\right) \cdot \left(2\right) - \left(- 5\right) \cdot \left(4\right)\right)$

=〈-22,64,34〉=vecc

Verification by doing 2 dot products

〈-22,64,34〉.〈7,4,-3〉=(-22)*(7)+(64)*(4)+(34)*(-3)=0

〈-22,64,34〉.〈-5,2,-7〉=(-22)*(-5)+(64)*(2)+(34)*(-7)=0

So,

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