If one wishes to formalize the notion of vectors in space it is necessary to introduce an equivalence relation in the three dimensional space $\mathbb E^3$ of euclidean geometry which identifies oriented line segments which have the same length (and more).
Since one mentions ``length'' in that equivalence relation, it is implicit we are using some fixed metric in $\mathbb E^3$, right? I'm asking because all the books I've seen on analytic geometry does not mention this. So, is the metric really there or is there some geometric trick to avoid talking about it?
Is there any reference which discuss this kind of formalism?
Thanks.
P.s.: I see $\mathbb E^3$ as an abstract set from which one distinguishes the subsets whose elements are lines and whose elements are planes. So, I initially don't have a coordinate system, for instance.