Why do we need position vectors? Position vector is a vector which defines location. It is also the only kind of vector which can't be moved. But coordinates can also do the same work. They also define position. So why do we need position vector? Is it because subtraction of position vectors results in displacement vector and subtraction of two points can not give a vector?
Edit: I think that I have found the answer. But I'm not sure if it is right or not. This is what I thought
When we change frame of reference and we want to determine the position relative to changed frame of reference , then I can just add the position vectors of first frame of reference respect to second frame of reference and that of object with respect to first frame of reference to get the position vector of object with respect to second frame of reference. But with points I can not do such a thing.
 A: Position vectors are relative to an origin (which may itself be moving e.g. position relative to a point on the surface of the Earth) but are independent of the co-ordinate system used. They therefore provide a convenient way of describing relative positions. We can say that the displacement of object 2 relative to object 1 is always $\vec r_2 -\vec r_1$. Trying to express this displacement in terms of the spherical co-ordinates of the two objects relative to some third point is quite complicated.
A: 
So why do we need position vector?

We do not need the position vector, it is actually somewhat artificial. If it were a vector then there would necessarily be a naturally meaningful sense of vector addition for positions. For example, Paris has a position $r_P$ and London has a position $r_L$. If position were naturally a vector then $\vec r_P + \vec r_L$ would naturally be a position also, but it isn't without some additional specifications.
In flat spacetime, position is much better treated as an affine space. An affine space is sometimes loosely described as a vector space that forgot about its origin. In an affine space there is no addition operation, but there is a subtraction operation. The subtraction of two elements of an affine space is a vector. So, in our previous example (ignoring the curvature) while $\vec r_P + \vec r_L$ is not meaningful, $r_P - r_L$ naturally gives a displacement vector. This vector is the one specific direction and one specific distance you need in order to go from London to Paris.
However, when you are dealing with curved spacetime, position is not even an affine space. To see why, consider a sphere. If you take the position of the north pole $r_N$ and the position of the south pole $r_S$ then the difference between them $r_N-r_S$ is not unique. You can go any direction from the south pole and reach the north pole simply by going the right distance.
So, in short, we do not need a position vector, and its usefulness as an auxiliary quantity diminishes as you get to more advanced physics.
A: The standard example of a position vector is a tangent vector. A vector that is a tangent vector must be tangent to something and this something is a curve on the surface or manifold in its direction at the point the vector is attached to.
Another standard example is an affine vector. An affine vector is a vector attached to a position (in flat space). And we can see from this that position vectors are more natural than free vectors since displacement, position and velocity vectors in Newtonian physics are affine vectors. Its also noticeable that Euclidean geometry is developed in affine rather than vector spaces.
Although affine space is more natural than vector spaces, its actually easier to develop the mathematics of vector spaces and then affine spaces, mainly because there is a simple axiomatic description of vector spaces but not of affine spaces.
A: In a co ordinate system, the position of a point is shown by some coordinates. If its a 2D system, the position will be shown by (x,y) and if the system is 3D then the vector will be (x,y,z). Now if these points are expressed in terms of xi+yj or xi+yj+zk, these vectors are position vectors.
Again position vectors lead to many important concepts such as triangle law of addition, parallelogram of addition and others which coordinates alone can't do.
For instance if I tell you to locate place on Earth physically and I give you a position vector( tail and head) and also the latitude, longitude which would you choose to use. Obviously position vector since the head of the vector gives you the answer. Latitude and Longitudes also give you the same result, but are quite complex and time consuming.
Hope this helps.
