Say for example v = <3,4>
I was taught to divide each component by the magnitude in order to get the unit vector, i.e.
3^2 + 4^2 = ||v||^2
5 = ||v||
So the unit vector of that vector is <3/5,4/5> or 1/5<3,4>
But if I forgot that I had to divide the components by the magnitude, I would not know how to get the unit vector. So I guess I am asking why you divide the components by the magnitude in order to get the unit vector?