Basically the answer to your question is that you are working in an Euclidean space. The vector mathematics used to work with vectors in Physics at this level tends to ignore the "point of attachment" of the vector in question. So we are free to move the vectors around. You tend to lose this ability when you start working with non-Euclidean Mathematics in, say, Relativity. Particularly in General Relativity.
Oh, another example, and one that's more on the level you are speaking of. It is convenient to remember what point the vectors are attached to when you are working in rotational dynamics. For Newton's Law problems we don't care where the forces are located, but when we start talking about torques we need to know what point the forces are acting at.