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#
