My question really is that simple. I am working in $\mathbb{R}^n$, I am inclined to believe the zero vector expressed on the basis of a proper subspace is distinct from that which is expressed on a basis of the orthogonal complement subspace. But all zero vectors of subspaces correspond with the origin of $\mathbb{R}^n$. So for example the $Z$ axis and the $X\times Y$ plane intersect at the origin of $\mathbb{R}^3$.
I don't recall ever having this clarified.
A subspace of $\mathbb{R}^n$ is literally just a subset of $\mathbb{R}^n$. So, its zero vector is literally just the zero vector $(0,0,\dots,0)\in\mathbb{R}^n$. In particular, every subspace of $\mathbb{R}^n$ has the same zero vector.