Why do we need to normalize vectors for?
Is it just to cut down on the arithmetic when finding other quantities of the vector?
Does it make life simpler to normalize vectors?
If you work with unit vectors only, then you take a whole dimension (magnitude) out of the problem; for example, the cross product of unit vectors is again a unit vector, the product of unit quaternions is again a unit quaternion, GL(n) deformation retracts onto O(n), and so on.