# What is the cross product of <4 , 5 ,-7 > and <5 ,1 ,-3 >?

May 18, 2018

The vector is =〈-8,-23,-21〉

#### 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,-7〉 and vecb=〈5,1,-3〉

Therefore,

$| \left(\vec{i} , \vec{j} , \vec{k}\right) , \left(4 , 5 , - 7\right) , \left(5 , 1 , - 3\right) |$

$= \vec{i} | \left(5 , - 7\right) , \left(1 , - 3\right) | - \vec{j} | \left(4 , - 7\right) , \left(5 , - 3\right) | + \vec{k} | \left(4 , 5\right) , \left(5 , 1\right) |$

$= \vec{i} \left(\left(5\right) \cdot \left(- 3\right) - \left(- 7\right) \cdot \left(1\right)\right) - \vec{j} \left(\left(4\right) \cdot \left(- 3\right) - \left(- 7\right) \cdot \left(5\right)\right) + \vec{k} \left(\left(4\right) \cdot \left(1\right) - \left(5\right) \cdot \left(5\right)\right)$

=〈-8,-23,-21〉=vecc

Verification by doing 2 dot products

〈-8,-23,-21〉.〈4,5,-7〉=(-8)*(4)+(-23)*(5)+(-21)*(-7)=0

〈-8,-23,-21〉.〈5,1,-3〉=(-8)*(5)+(-23)*(1)+(-21)*(-3)=0

So,

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