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.
Why do we normalize certain vectors?
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soft-question
vectors
riemannian-geometry
1 Answers
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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 $$\langle u, v \rangle\cdot v $$
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1Ah yes, maybe you could add the Gram-Schmidt algorithm as an example for your second point. Thanks. – 2017-01-10
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0@TheGeekGreek I included the most general example I could come up with: projecting a vector onto another. In Gram-Schmidt you just project a bunch of vectors onto eachother – 2017-01-10