As I understand it, the correspondence between the zero vector and the origin (0,0,...,0) in n-dimensional space arises when realising an abstract vector space with a coordinate system - in your case R^n.

It is useful to think intuitively of the zero vector as the point at the origin, and the distinction between them is blurred when talking about R^n. However it is when you start talking about abstract vector spaces that the notion of the zero vector as a point is no longer applicable, because "points" as you imagine them in R^n do not really exist any longer.