TITLE: why don't we define vector multiplication component-wise?
QUESTION [38 upvotes]: I was just wondering why we don't ever define multiplication of vectors as individual component multiplication.  That is, why doesn't anybody ever define $\langle a_1,b_1 \rangle \cdot \langle a_2,b_2 \rangle$ to be $\langle a_1a_2, b_1b_2 \rangle$?  Is the resulting vector just not geometrically interesting?

REPLY [18 votes]: To elaborate yet a bit more on what Guiseppe Negro, James S. Cook and Michael Joyce have already said:

A vector is not a tuple of individual components. A vector is an element of some vector space.

When you're writing a vector as such a tuple, you're only referring to the expansion of the vector in some particular basis. But this basis is very often not even specified. Which is actually ok, because the "normal" vector operations don't in fact depend on the choice, i.e. if you transformed all you vectors to be written out in some other basis, you would have all different numbers but the same calculations would still yield correct results.
But that wouldn't work for component-wise multiplication, as was already shown. This operation simply does not work on the vectors but on their basis representation, which is only well-defined for some fixed choice of basis, which is not what you're actually interested in when studying vectors.
Of course, there are plenty of applications where you are in fact interested in tuples of numbers, but those aren't vectors then. There's nothing wrong with the Hadamard product, but it doesn't work on vectors but on matrices1. If you want to multiply components, then your objects may be called tuples or arrays or lists or whatever, but hardly vectors.
Unfortunately, many people have got this wrong, and that's why e.g. C++ programmers are blessed with2 an std::vector class that is in fact for dynamic arrays, which are even less accurately vectors than static arrays.

1Matrices suffer from a similar problem: many people use "matrix" and "linear mapping" as synonyms, but they aren't in fact the same; matrices refer to a particular basis while linear mappings need no such thing.
2Nothing against std::vector, it's great – it's just not a vector class, just like "functors" aren't in fact functors.