Is this Vector operation defined? Does it have a name? Let's say I have 2 vectors:
[a, b, c]
[x, y, z]

And I need to do an operation like the following for a computer program:
[a*x, b*y, c*z]

What I want to know is if this operation is defined as a common vector operation? And what can I call it? I ask this because I can't find any information about it. Of course nothing tells me I can't do this, it may just not be a "normal" vector operation.
 A: Your operation is the multiplication in the algebra of functions $\{1,\ldots,n\}\to\mathbb R$.
This algebra is a vector space with the obvious pointwise addition and scalar multiplication. It is isomorphic as a vector space to the usual $n$-dimensional vector space $\mathbb R^n$ (depending on the details of the formalization you work in, they may even be identical rather than simply isomorphic). Nevertheless, it is most common to think of it as something separate from $\mathbb R^n$.
Why is that? It is because usually when we speak about $\mathbb R^n$ we want to think of it as a concrete representation of some other vector space which we get by choosing a basis, and in most cases we have considerable liberty to choose the basis. Therefore $\mathbb R^n$ usually invokes an idea of "some abstract $n$-dimensional vector space with a more or less arbitrarily chosen basis".
However, your multiplication operation does not fit into this framework, because it is not preserved when we move to another basis. Therefore it is not a useful concept to speak of in the common situation where the basis was chosen more or less arbitrarily.
So in order to reduce the risk of confusion, we usually pretend that the component-wise product doesn't exist when we speak merely of $\mathbb R^n$. In the few situations where the multiplication is useful, we prefer to call the space something else such as to remind ourselves that our usual freedom to switch to another basis whenever it's convenient doesn't exist anymore.
In particular, you won't find the the Hadamard product listed in an article about "Euclidean vectors" -- because what makes the vectors Euclidean is that the relations between them do not change if we rotate the coordinate system. But the product does change if the coordinate system is turned.
A: Component-wise multiplication (or coordinate-wise multiplication like @Theo mentioned in his comment).
