If you have a vector (a,b,c) and multiply all the components by a same scalar alpha, you obtain the vector (alpha a, alpha b, alpha c). This vector is parallel to the original one, and it is smaller than the original if alpha<1, and greater than the original if alpha>1.
Now, let's talk about norms. The norm of a vector is the quantity
norm(a,b,c)=sqrt(a^2+b^2+c^2).
So, we want to multiply the vector (2,3,-2) by some scalar alpha such that the new vector (2\alpha,3\alpha,-2\alpha) has norm equal to one.
To do so, choose alpha as the inverse of the norm of the vector: we have
norm(2,3,-2)=sqrt(4+9+4)=sqrt(17)
and choosing alpha=1/sqrt(17) we have
norm(2\alpha,3\alpha,-2\alpha)=norm(2/sqrt(17), 3/sqrt(17),-2/sqrt(17))
= (2/sqrt(17))^2+(3/sqrt(17))^2+(-2/sqrt(17))^2
=4/17+9/17+4/17=1
