Confused about definition of three dimensional position operator in QM My QM text defines the position operator as follows:

The position operator $X= (X_1,X_2,X_3)$ is such that for $j=1,2,3:  \ X_j \psi(x,y,z)= x_j \psi(x,y,z)$.

To me this can mean two things. 
1) $X$ is a vector and acts as $X \psi(x,y,z)= (x \psi(x,y,z), y \psi(x,y,z), z \psi(x,y,z))$. But this doesn't make sense as $X$ is an observable/operator and so must send vectors to vectors (here functions).
2)There are three position operators $X_1, X_2, X_3$ and each act as defined.
How does the postion operator act on a state? Could anyone help me out here? Thanks!
 A: The Hilbert space can be seen as a direct product of three independent Hilbert spaces. 
$$|x,y,z\rangle=|x\rangle\otimes|y\rangle\otimes|z\rangle$$
When expressed as such, the position operator is also seen as a direct product of three operators each acting on their corresponding spaces. 
So technically they’re three eigenvalue equations combined into one. 
A: This is quite an odd way to introduce the position operator, I have to admit. Both definitions you have used are correct, they're just used in different ways in quantum mechanics.
In the the first one, $X$ is what is technically called a vector operators, in this case it's a little like a vector but the components are matrices (or operators). Sometimes it's useful to do this and we can do a sort of dot-product with other vector operators (which you will probably come across soon in QM).
$X$ is composed of the three operators you've defined in 2), and when we want to think about the position operator in three dimensions, definition 1) does actually work. It's a little bit odd, but like I said, $X$ isn't an operator, $X$ is a vector operator, and so the technical issue of mapping the wavefunction to a vector isn't actually an issue. If it's still puzzling, then you can try to think about it as each component of the vector $X\psi$ as being an individual function, and then noticing that this is really just a way of putting three separate scalar equations into one vector equation.
A: The position vector operator, $X=(X_1,X_2,X_3)$, is usually just used as shorthand for writing things like $P\cdot X$, which is defined to be $X_1P_1+X_2P_2+X_3P_3$. You are right that it is not an operator that acts on the Hilbert space of wave functions.
