Why do we need to normalize vectors for?Is it just to cut down on

matqkks

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?

micromass

Hi matqkks!

It is sometimes essential. For example, the Fourier series w.r.t. a orthogonal basis is given by

[tex]\sum_{n=0}^{+\infty}{<x,e_i>\frac{e_i}{\|e_i\|}}[/tex]

This formula wouldn't be true if we didn't normalize the e_i.

But most of the time, I guess it's just easier. You can perform a Gramm-Schmidt procedure where you just obtain an orthogonal sequence, but it's nicer if you also know it's orthonormal. It makes a lot of formula's easier if you normalize.

chiro

There are a variety of reasons why you might want to normalize a vector.

One reason includes projections. Another reason might include the need to find angles between vectors: in order to do this you need to normalize vectors so the a . b = |a| |b| cos(a,b) and since |a| = |b| = 1, you can get cos(a,b) directly.

Also things like decomposition require this. As micromass has pointed out above, decompositions require that you normalize elements.

When you are doing projections in integral transforms, you are dealing with orthonormal basis and one property of orthonormal basis is that the the length of a basis vector is unit length (i.e. 1). If you ever look at integral transforms like fourier transforms and wavelet transforms, you will see what I am talking about.