# What is the cross product of <5, -3,1 > and <-5 ,2 ,3 >?

Dec 21, 2016

The answer is =〈-11,-20,-5〉

#### Explanation:

The cross product is obtained from the determinant

$| \left(\hat{i} , \hat{j} , \hat{k}\right) , \left(d , e , f\right) , \left(g , h , i\right) |$

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

Therefore,

$| \left(\hat{i} , \hat{j} , \hat{k}\right) , \left(5 , - 3 , 1\right) , \left(- 5 , 2 , 3\right) |$

$= \hat{i} | \left(- 3 , 1\right) , \left(2 , 3\right) | - \hat{j} | \left(5 , 1\right) , \left(- 5 , 3\right) | + \hat{k} | \left(5 , - 3\right) , \left(- 5 , 2\right) |$

$= \hat{i} \left(- 3 \cdot 3 - 2 \cdot 1\right) - \hat{j} \left(5 \cdot 3 + 5 \cdot 1\right) + \hat{k} \left(5 \cdot 2 - 5 \cdot 3\right)$

$= \hat{i} \left(- 11\right) - \hat{j} \left(\right) 20 + \hat{k} \left(- 5\right)$

=〈-11,-20,-5〉

Verification , by doing a dot product

〈-11,-20,-5〉.〈5,-3,1〉=(-55+60-5)=0

〈-11,-20,-5〉.〈-5,2,3〉=(55-40-15)=0