It might be helpful to remember that the definitions of R2 and R3 are not geometric at all. R2 is the set of all ordered pairs of real numbers, whereas R3 is the set of all ordered triples of real numbers. W is a subspace fo R3, and so it still consists of elements which are triples, not pairs.
The fact that R2 and W can be visualized with the same geometric picture, namely the xy plane, is one way to see in a concrete way the isomorphism which the second poster refered to.
