First, one must understand that a vector is in some sense a hybrid object. Having both length and direction is not strictly a set. We use triples of real numbers to represent vectors and endow them with length and direction.
In axiomatic geometry we define an angle as the union of two rays which have a common end point. We can extend that idea to vectors with the above understanding. In that understanding all vectors can be seen as having the origin as a common endpoint. Thus, any two vector have an angle between them measuring from to .
Does that description help you at all?