# What is the cross product of <10 ,4 ,1 > and <-5 ,2 ,3 >?

Mar 2, 2018

The vector is =〈10,-35,40〉

#### 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=〈10,4,1〉 and vecb=〈-5,2,3〉

Therefore,

$| \left(\vec{i} , \vec{j} , \vec{k}\right) , \left(10 , 4 , 1\right) , \left(- 5 , 2 , 3\right) |$

$= \vec{i} | \left(4 , 1\right) , \left(2 , 3\right) | - \vec{j} | \left(10 , 1\right) , \left(- 5 , 3\right) | + \vec{k} | \left(10 , 4\right) , \left(- 5 , 2\right) |$

$= \vec{i} \left(\left(4\right) \cdot \left(3\right) - \left(1\right) \cdot \left(2\right)\right) - \vec{j} \left(\left(10\right) \cdot \left(3\right) + \left(5\right) \cdot \left(1\right)\right) + \vec{k} \left(\left(10\right) \cdot \left(2\right) - \left(4\right) \cdot \left(- 5\right)\right)$

=〈10,-35,40〉=vecc

Verification by performing 2 dot products

〈10,4,1〉.〈10,-35,40〉=(10)*(10)+(4)*(-35)+(1)*(40)=0

〈-5,2,3〉.〈10,-35,40〉=(-5)*(10)+(2)*(-35)+(3)*(40)=0

So,

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