# What is the cross product of [1, -2, -3] and [3, 7, 9]?

Mar 24, 2018

The vector is =〈3,-18,13〉

#### 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=〈1,-2,-3〉 and vecb=〈3,7,9〉

Therefore,

$| \left(\vec{i} , \vec{j} , \vec{k}\right) , \left(1 , - 2 , - 3\right) , \left(3 , 7 , 9\right) |$

$= \vec{i} | \left(- 2 , - 3\right) , \left(7 , 9\right) | - \vec{j} | \left(1 , - 3\right) , \left(3 , 9\right) | + \vec{k} | \left(1 , - 2\right) , \left(3 , 7\right) |$

$= \vec{i} \left(\left(- 2\right) \cdot \left(9\right) - \left(- 3\right) \cdot \left(7\right)\right) - \vec{j} \left(\left(1\right) \cdot \left(9\right) - \left(- 3\right) \cdot \left(3\right)\right) + \vec{k} \left(\left(1\right) \cdot \left(7\right) - \left(- 2\right) \cdot \left(3\right)\right)$

=〈3,-18,13〉=vecc

Verification by doing 2 dot products

〈3,-18,13〉.〈1,-2,-3〉=(3)*(1)+(-18)*(-2)+(13)*(-3)=0

〈3,-18,13〉.〈3,7,9〉=(3)*(3)+(-18)*(7)+(13)*(9)=0

So,

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