Why can we translate vectors freely in space? This question has been bugging me for a while. Assume the following set up:



*

*Consider a 2-dimensional world.

*I have a ball at my feet. Assume the ball is at O(0, 0). 

*My friend and I both kick the ball at the same time. I exert a force V1 and my friend exerts V2.

*The ball should travel along resultant of V1 and V2, say R.

*Suppose I translate V1 along an axis perpendicular to it, to V1' say.

*Now if V1' and V2 alone act on the ball at the same time, it will move along V2 alone as V1' acts on a different point in space after translating it.


Doesn't this mean that translating a vector changes it? Clearly V1 and V1' result in different outcomes. Shouldn't a vector be represented by where its tail lies in a frame of reference?
I have come across the following statement quite often: "In Physics, vectors can freely translate in space without changing". What is the meaning of this statement? Maybe I am confused by what a vector itself actually means. Where is the error in my understanding above? Forces are represented by vectors right? Do they belong to a subclass of a more general vector class? Any help appreciated. Thanks in advance.
 A: Your confusion is caused by the fact that you were never taught the distinction between a vector space and an affine space.
The difference between a 1 dimensional vector space and a line, is that on a line, all points are equivalent. There is no “distinguished point”. When you choose an origin on a line, a completely arbitrary decision, you make your line correspond to a 1 dimensional vector space. If you then choose a basis for it, every vector is just a scalar multiple of that one basis element. This is how you get a number line.
Similarly, the difference between a two dimensional vector space and a plane is that on a plane, all points are equivalent, Again, there is no “distinguished point”. When you choose an origin on a plane, a completely arbitrary decision, you make your plane correspond to a 2 dimensional vector space. If you choose a basis, it has 2 elements, and so that 2 dimensional vector space becomes a Cartesian product of 2 scalars, which is how you get the familiar plane graph.
What is this correspondence? The axioms of geometry include that the translations act simply transitively, which means that given any two points, there is a unique translation taking one to the other. So when you draw a vector from p to q, you are describing the translation which takes p to q. When you pick an origin, each point is associated with the translation taking the origin to that point.
So in one picture you are focusing on the affine space, in the other picture you are focusing on the vector space.
Every geometry satisfying the axiom of Desargues has a group of translations which form a vector space over a division ring, and every vector space over a division ring gives rise to such a geometry. The geometric (affine) picture and the algebraic (vector space) picture are completely equivalent.
