# What is the cross product of [1, -2, -3] and [2, -5, 8] ?

Dec 7, 2016

The answer is =〈-31,-14,-1〉

#### Explanation:

The cross product of 2 vectors

veca=〈a_1,a_2,a_3〉

and vecb=〈b_1,b_2b_3〉

is given by

the determinant $| \left(\hat{i} , \hat{j} , \hat{k}\right) , \left({a}_{1} , {a}_{2} , {a}_{3}\right) , \left({b}_{1} , {b}_{2} , {b}_{3}\right) |$

$= \hat{i} \left({a}_{2} {b}_{3} - {a}_{3} {b}_{2}\right) - \hat{j} \left({a}_{1} {b}_{3} - {a}_{3} {b}_{1}\right) + \hat{k} \left({a}_{1} {b}_{2} - {a}_{2} {b}_{1}\right)$

Here we have,

〈1.-2-3〉 and 〈2,-5,8〉

So, the cross product is

$| \left(\hat{i} , \hat{j} , \hat{k}\right) , \left(1 , - 2 , - 3\right) , \left(2 , - 5 , 8\right) |$

$= \hat{i} \left(- 16 - 15\right) - \hat{j} \left(8 + 6\right) + \hat{k} \left(- 5 + 4\right)$

=〈-31,-14,-1〉

Verification (the dot product of perpendicular vectors is $= 0$)

〈-31,-14,-1〉.〈1.-2-3〉=-31+28+3=0

〈-31,-14,-1〉.〈2,-5,8〉=-62+70-8=0