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

Jan 5, 2017

The answer is =〈-60,60,20〉

#### Explanation:

The cross product of 2 vectors, 〈a,b,c〉 and d,e,f〉

is given by the determinant

$| \left(\hat{i} , \hat{j} , \hat{k}\right) , \left(a , b , c\right) , \left(d , e , f\right) |$

$= \hat{i} | \left(b , c\right) , \left(e , f\right) | - \hat{j} | \left(a , c\right) , \left(d , f\right) | + \hat{k} | \left(a , b\right) , \left(d , e\right) |$

and $| \left(a , b\right) , \left(c , d\right) | = a d - b c$

Here, the 2 vectors are 〈7,5,6〉 and 〈3,5,-6〉

And the cross product is

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

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

$= \hat{i} \left(- 30 - 30\right) - \hat{i} \left(- 42 - 18\right) + \hat{k} \left(35 - 15\right)$

=〈-60,60,20〉

Verification, by doing the dot product

〈-60,60,20〉.〈7,5,6〉=-420+300+120=0

〈-60,60,20〉.〈3,5,-6〉=-180+300-120=0

Therefore, the vector is perpendicular to the other 2 vectors