Even if the vectors don't originate at the origin you can express them as if they do. For the sake of my diagram, let go back to two dimensions for a bit:
Consider this situation, we have a vector AB and we want to represent it in terms of vectors from the origin, the following property of vectors can be used:
So back to three dimensions:
Where AB is a direction vector from the point A to the point B and x(a), y(a), z(a) etc. are values for the direction vector from the origin to the points A and B ( therefore also the position vectors of A and B ).
I hope this is what you were asking. =)