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Regarding Joel's question:
Where does a position vector reside?
First of all, it doesn't make much difference how you
conceive of or draw
the vectors when the manifold of interest is globally flat, since the
manifold itself and its tangent space to it at each point are
isomorphic.
In such a case we can treat vectors in the usual way as
having them reside
in the space of interest (i.e. R^3).
Regarding the specific question above, in the general curved
space case,
the position of a point in that curved manifold is *not* a
vector at all.
It is a point in the manifold. The points in a manifold are not
individually vectors. Vectors live in vector spaces; curved manifolds
are not vector spaces. Since the position is not a vector, it doesn't
reside as a vector anywhere. It just resides as a point just where it
is in the manifold.
*But*, the *differential displacement* along a path between two
infinitesimally nearby points in the manifold *is* a vector,
and it lives
in the tangent (vector) space of the point where that differential is
taken.