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

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?

• This is a very unusual terminology: What does the "root" of a vector mean? – Moishe Kohan Mar 5 '18 at 2:04
• I've edited the post, sorry for my terrible english. – PHT Mar 5 '18 at 2:15
• @PHT Please remember that you can choose an aswer among the given if the OP is solved, more details here meta.stackexchange.com/questions/5234/… – user Mar 9 '18 at 23:05

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.

• I think you mean "$AD$" and "$BC$," no? – Noah Schweber Mar 5 '18 at 2:10
• @NoahSchweber: No, but I had a lapsus calami anyway: I meant AC and $BD$ (the diagonals). Thanks for pointing it! – Bernard Mar 5 '18 at 2:14
• Really? Then $(A, B)\not\sim (A, B)$ if $A\not=B$ (unless I misunderstand what you mean by "midpoint"); I think $AD$ and $BC$ are the diagonals ... – Noah Schweber Mar 5 '18 at 2:35
• @NoahSchweber: You're right. For my excuse, it was late here when I posted… Thanks! I'll fix it at once. – Bernard Mar 5 '18 at 8:57

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 $(P,\vec v)$.

• Yes, that's it. Sorry for my english. – PHT Mar 5 '18 at 2:15