Why do we normalize certain vectors?

Question

In Riemnannian geometry for example one defines the unit normal of a local parametrization (under the right circumstances) as $$N := \frac{\partial_1X \times \partial_2X}{|\partial_1X \times \partial_2X|}$$ As the name says, it is a unit vector. So the other day I received a funny question, which I was not immediately able to answer: Why do we normalize this vector? Or more generally why it is desirable to normalize vectors? I mean, one advantage of normalizing basis vectors is that we can easily calculate the coefficients in the linear combinations, but I think there are more.

Answer 1

My two guesses are:

• In the most general way possible (not talking about a particular application in which you normalize vectors) it is because it standardizes calculations and theorems and what-not;

• Projections onto unitary vectors are cleaner (as in less calculations) than projections onto non-unitary vectors.

(Recall that the projection of $u $ over $v $ is

$$\frac{\langle u, v \rangle}{||v||}\cdot v $$

And if $v $ is normalized, then it simply becomes $$\langle u, v \rangle\cdot v $$