Why do we need a Unit Vector (in other words, why do we need to normalize vectors)?

Question

I am reading a book on game AI.

One of the terms that is being used is to normalize a vector which is to turn a vector into a unit. To do so you must divide each dimension x, y and z by its magnitude.

We must turn vector into a unit before we do anything with it. Why?

And could anyone give some scenarios where we must use a unit vector?

Thanks!

Answer 1

You don't have to normalize vectors, but it makes a lot of equations a little simpler when you do. It could also make API's smaller: any form of standardization has the potential to reduce the number of functions necessary.

Here's a simple example. Suppose you want to find the angle between two vectors u and v. If they are unit vectors, the angle is just arccos(uv). If they're not unit vectors, the angle is arccos(uv/(|u| |v|)). In that case, you end up computing the norms of u and v anyway.