# What is the cross product of [2, 4, 5] and [2, -5, 8] ?

Apr 5, 2018

The vector is =〈57,-6,-18〉

#### 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=〈2,4,5〉 and vecb=〈2,-5,8〉

Therefore,

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

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

$= \vec{i} \left(\left(4\right) \cdot \left(8\right) - \left(5\right) \cdot \left(- 5\right)\right) - \vec{j} \left(\left(1\right) \cdot \left(3\right) - \left(1\right) \cdot \left(1\right)\right) + \vec{k} \left(\left(- 1\right) \cdot \left(1\right) - \left(2\right) \cdot \left(1\right)\right)$

=〈57,-6,-18〉=vecc

Verification by doing 2 dot products

〈57,-6,-18〉.〈2,4,5〉=(57)*(2)+(-6)*(4)+(-18)*(5)=0

〈57,-6,-18〉.〈2,-5,8〉=(57)*(2)+(-6)*(-5)+(-18)*(8)=0

So,

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