Why do all vectors in vector space must have the same tail?

Question

In my high school, I learned all vectors that have same magnitude and direction should be treated in the same way. And I've learned abstract algebra for a while, and I am learning Vector space now. I do know what a vector space is, and my teacher let me think it as a set of vectors that have the same tail at the origin. We don't have vectors that have different tail. In my opinion, this is because a vector space is a group, so it has one unique identity element, and vectors should have one unique inverse. Is it right? Do you have other reasons why these vectors should have same tail?

Answer 1

Actually, vectors of geometry are the equivalence classes of pairs of points under the equipollence relation: $$(A,B)\sim (C,D) \iff [AD\mkern1.5mu]\enspace\text{and}\enspace[BC\mkern 1.5mu]\enspace\text{have the same midpoint}\qquad\text{(parallelogram law)}.$$ An equivalence class may be represented by any of its elements. Usually, one chooses the element starting at the origin.

Answer 2

Maybe the term root means tail?

In this case, when we deal with vector space, the vectors $\vec v$ are to be considered with the tail in the origin which corresponds to the zero vector.

In other context, notably in physics, we can also deal with vectors applied to a particular point $P$, in this case vectors are denote by $(P,\vec v)$.