TITLE: English names for vector beginning and end
QUESTION [12 upvotes]: I've done some research, but since English is not my native language, I'm struggling to find an answer to this:
Given a vector, what do you call its beginning and end points?
The best I've found so far is the word "base" for the beginning point of a vector, but I have no clue if that's correct. The best I've got for the end point is the "tip" of the vector, although the same applies as before.

REPLY [5 votes]: The Wikipedia article Euclidean vector says that when you construct a vector, called $\overrightarrow{AB}$, from two points $A$ and $B$ in Euclidean space, then $A$ is called initial point, and $B$ is called terminal point.
It all depends on what your definition of a vector is, of course. For example, it is common to consider a vector as an equivalence class of all those oriented line segments of this kind that have the same length (magnitude) and the same direction. With such a definition, a "vector" has no initial point and no terminal point, although you can pick any initial point you like and consider the representative oriented line segment originating from that point.
In the more general setting of a smooth manifold, you often have a tangent vector in the particular tangent space (or fiber) $T_pM$ that corresponds to a point $p$ in the manifold. In such a case, you need both this foot point (is this the common name?) $p$ and a representation of the vector in the tangent space "sitting" at that point. In this general setting, there is no canonical equivalence between a vector in $T_pM$ and a vector in another fiber $T_qM$ (here $q\ne p$ is another point in the manifold $M$).
Usually, when $v\in T_pM$, we just say that $v$ is a tangent vector "at" $p$. As I said, I think I heard $p$ being called the foot point of $v$.