# What is the cross product of <7, 2 ,6 > and <-3 ,5 ,-6 >?

Jun 26, 2018

The vector is =〈-42,24,41〉

#### 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=〈7,2,6〉 and vecb=〈-3,5,-6〉

Therefore,

$| \left(\vec{i} , \vec{j} , \vec{k}\right) , \left(7 , 2 , 6\right) , \left(- 3 , 5 , - 6\right) |$

$= \vec{i} | \left(2 , 6\right) , \left(5 , - 6\right) | - \vec{j} | \left(7 , 6\right) , \left(- 3 , - 6\right) | + \vec{k} | \left(7 , 2\right) , \left(- 3 , 5\right) |$

$= \vec{i} \left(\left(2\right) \cdot \left(- 6\right) - \left(6\right) \cdot \left(5\right)\right) - \vec{j} \left(\left(7\right) \cdot \left(- 6\right) - \left(6\right) \cdot \left(- 3\right)\right) + \vec{k} \left(\left(7\right) \cdot \left(5\right) - \left(2\right) \cdot \left(- 3\right)\right)$

=〈-42,24,41〉=vecc

Verification by doing 2 dot products

〈-42,24,41〉.〈7,2,6〉=(-42)*(7)+(24)*(2)+(41)*(6)=0

〈-42,24,41〉.〈-3,5,-6〉=(-42)*(-3)+(24)*(5)+(41)*(-6)=0

So,

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