It's for the simple reason that when we "count" we need a unit. If we are "counting" vectors, we need to know the "unit" vector, so that we know what "1" is.
I have a problem in understanding the conceptual meaning of normalization. I know that normalization is essentially to make an coefficient or vectors equaled to 1. However, I don't quite understand why we need normalization, the rules for normalization and the interpretation of normalization .
Can people help me to clarify my thought?
normal form n. a canonical form to which a given structure or object may be reduced.
normalize vb. to put into some normal form.
Now, of course, "canonical form" is probably not any less in need of elaboration than "normal form".
But the general idea behind "normal forms" is this: you have certain "mathematical objects" of a rather abstract nature, like functions, or direction in a space, or .. you name it.
And you have certain syntactical representations of those mathematical objects. Ideally, the syntactical representation would be unique, so that one could decide about the equality or inequality of the mathematical objects represented by the two syntactical forms by just comparing the syntactical forms themselves. For example, you are given two functions, each defined by some rather elaborate term, and you are asked to check whether these two terms actually represent/define the same function or not.
Similarly, a normalized vector is a unique representation of a particular direction in a given vector space. If you want to check whether two given vectors have the same direction, you might usefully normalize both of them (because this does not change their direction) in order to do so.