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Distinction between vectors and points
I have a doubt about the distinction between points and vectors. I know there's already a topic about that here in the web site, but i thought the correct was to create a new one. Well, the question is: in euclidean space we identify both points and vectors with elements of $\mathbb{R}^n$, but I know they're different things.
And I know that when dealing with general manifolds the situation gets worse and it's needed to define precisely the notion of a tangent space at each point of the manifold. So my question is: how is it possible to define precisely the distinction between points and vectors first in euclidean space and then in general manifolds ?
I've seem a book on differential geometry where the author introduces the operation of addition of points and multiplication of point by scalar, but i did think that these operations are meaningless geometrically speaking.
I've heard about the notion of an affine space, is that the correct way to make a rigorous distinction between vectors and points?
Thanks in advance for the help.