- Conceptually, a vector is a direction and magnitude. When moving, a particle has a speed, and a direction in which they are moving. But this is a concept, not a definition.
- The mathematical definition of a vector is any element of a vector space. Where a "vector space" is defined as a set $V$ with an particular element called "$0$" and two binary operations called addition and scalar multiplication that satisfy certain properties that you can find listed in many places.
$\Bbb R^n$ is easily seen to be a vector space, where $0$ is the element $(0,\dots, 0)$, addition is just adding by coordinates: $(x_1, \dots, x_n) + (y_1, \dots, y_n) = (x_1 + y_1, \dots, x_n + y_n)$ and scalar multiplication is multiplying each coordinate: $r(x_1, \dots x_n) = (rx_1,\dots, rx_n)$ for $r \in \Bbb R$.
$\Bbb R^n$ also serves as a model for $n$-dimensional Euclidean space. This gives us two different ways to view an element of $\Bbb R^n$:
- It can be considered a point in Euclidean space. Points are simply places where things can be. They are positions in space. Here we ignore the role of $0$, and the existence of addition and scalar multiplication.
- It can be considered a vector - not a position, but a direction and a magnitude, which the magnitude respresenting how strong some action is associated with that direction. Here we are ignoring position. A (mathematical) vector doesn't know where it is at - only in what direction something is acting, and how strong that action is.
Physicists prefer to combine these two concepts. For them, a vector is not only the direction and magnitude of an action, but also has the position where the action occurs. You can only add these physics vectors if they have the same position. Vectors at different points cannot be combined. But if you collect all the possible vectors at a given point, they can all be added together, and multiplied by scalars, and thus this collections of all vectors at a point forms a mathematical vector space, called the tangent space at that point. If you move to a different point and collect all its possible vectors, you get another tangent space at this new point. The vectors at this point are entirely different from the vectors at the first point. The two tangent spaces have no vectors in common.
This is independent of the shape of space. Just like the plane, the sphere consists of points, and at each point in it, there are various different directions in which one can travel, and rates at which you can travel in those directions. You can't add points on the sphere to each other to get other points on the sphere. But you can combine directions of travel on the sphere just like you do in the plane. Thus each point on the sphere has its own tangent space, just like points on the plane. While the sphere curves, there is no such concept for vectors.
The tangent space at a point on a sphere is just a flat vector space. We can identify it with the plane in space tangent to the sphere at the point (thus the name "tangent space"). But that identification is not fully accurate, as the planes tangent to the sphere at various points intersect each other, while the tangent spaces are always disjoint from each other.
When the space is a nice flat Euclidean space, we can choose coordinates on it and extend them to the entire space, to treat the space as $\Bbb R^n$. Then we can use the vector space structure on $\Bbb R^n$ to treat the points of the Euclidean space as if they were vectors themselves. When you do this, you find that you can identify the tangent space at each point with $\Bbb R^n$ itself, and instead of dealing with different vector spaces at each point, you can just use this identification to divorce the physics vector from its point, so it only has direction again, and all points can now share the same vector space.
When you do this, you treat points as vectors, and the position - the point where the action is taking place - is then called the "position vector".
But whent the space is not Euclidean, this is not possible. Such spaces do not have "position vectors".
