# What is the cross product of <4 , 5 ,-9 > and <4, 8 ,-2 >?

Dec 27, 2016

The answer is =〈62,-28,12〉

#### 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 〈4,5,-9〉 and 〈4,8,-2〉

And the cross product is

$| \left(\hat{i} , \hat{j} , \hat{k}\right) , \left(4 , 5 , - 9\right) , \left(4 , 8 , - 2\right) |$

$= \hat{i} | \left(5 , - 9\right) , \left(8 , - 2\right) | - \hat{j} | \left(4 , - 9\right) , \left(4 , - 2\right) | + \hat{k} | \left(4 , 5\right) , \left(4 , 8\right) |$

$= \hat{i} \left(- 10 + 72\right) - \hat{i} \left(- 8 + 36\right) + \hat{k} \left(32 - 20\right)$

=〈62,-28,12〉

Verification, by doing the dot product

〈62,-28,12〉.〈4,5,-9〉=248-140-108=0

〈62,-28,12〉.〈4,8,-2〉=248-224-24=0

Therefore, the vector is perpendicular to the other 2 vectors