What is the cross product of <9,2,8 > and <6, -2, 7 >?

Mar 6, 2018

The vector is $= < 30 , - 15 , - 30 >$

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=〈9,2,8〉 and vecb=〈6,-2,7〉

Therefore,

$| \left(\vec{i} , \vec{j} , \vec{k}\right) , \left(9 , 2 , 8\right) , \left(6 , - 2 , 7\right) |$

$= \vec{i} | \left(2 , 8\right) , \left(- 2 , 7\right) | - \vec{j} | \left(9 , 8\right) , \left(6 , 7\right) | + \vec{k} | \left(9 , 2\right) , \left(6 , - 2\right) |$

$= \vec{i} \left(\left(2\right) \cdot \left(7\right) - \left(8\right) \cdot \left(- 2\right)\right) - \vec{j} \left(\left(9\right) \cdot \left(7\right) - \left(8\right) \cdot \left(6\right)\right) + \vec{k} \left(\left(9\right) \cdot \left(- 2\right) - \left(2\right) \cdot \left(6\right)\right)$

=〈30,-15,-30〉=vecc

Verification by doing 2 dot products

〈30,-15,-30〉.〈9,2,8〉=(30)*(9)+(-15)*(2)+(-30)*(8)=0

〈30,-15,-30〉.〈6,-2,7〉=(30)*(6)+(-15)*(-2)+(-30)*(7)=0

So,

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