# What is the cross product of [3,-1,2] and [-2,0,3] ?

Nov 5, 2016

The cross product is =〈-3,-13,-2〉

#### Explanation:

The cross product of two vectors vecu=〈u_1,u_2,u_3〉
and vecv=〈v_1,v_2,v_3〉 is the determinant
∣((veci,vecj,veck),(u_1,u_2,u_3),(v_1,v_2,v_3))∣

=$\vec{i} \left({u}_{2} {v}_{3} - {u}_{3} {v}_{2}\right) - \vec{j} \left({u}_{1} {v}_{3} - {u}_{3} {v}_{1}\right) + \vec{k} \left({u}_{1} {v}_{2} - {u}_{2} {v}_{1}\right)$

Here we have vecu=〈3,-1,2〉 and vecv=〈-2,0,3〉

So the cross product is vecw=〈veci(-3)-vecj(-13)+veck(-2〉
=〈-3,-13,-2〉
To check, we verify that the dot products are $= 0$
$\vec{w} . \vec{u} = \left(- 9 + 13 - 4\right) = 0$
$\vec{w} . \vec{v} = \left(6 + 0 - 6\right) = 0$