Answer
$\frac{\sqrt {19}}{19}i+\frac{3\sqrt {19}}{19}j+\frac{3\sqrt {19}}{19}k$
Work Step by Step
1. Given $\vec v=< 3, 1, -2>$ and $\vec w=<-3,2,-1>$, we have
$\vec v\times\vec w=<(1(-1)-(-2)2),-(3(-1)-(-2)(-3)),(3(2)-1(-3))>=3i+9j+9k=3(i+3j+3k)$
2. As $||\vec v\times\vec w || =3\sqrt {1+9+9}=3\sqrt {19}$, we can find the required unit vector as
$\frac{3(i+3j+3k)}{3\sqrt {19}}=\frac{\sqrt {19}}{19}i+\frac{3\sqrt {19}}{19}j+\frac{3\sqrt {19}}{19}k$
(the opposite direction will also be true.)
