English names for vector beginning and end 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.
A good question. Since an important property of a vector is its direction it is hard to talk about vectors without having words for where they start and end.  

In my experience we have generally called the source or beginning of a vector its "tail" and the destination or end of the vector its "head".
Uh, you aren't actually talking about a vector here.  A vector is not defined by a beginning and end point.  It has a direction and magnitude (and equivalents in other coordinate systems) but no fixed origin.

The vector might have a location where the coordinates are taken.  This is particularly relevant in curvilinear coordinate systems.  If you draw a vector there, you typically draw the respective arrow with its base on the location of the vector, its shaft being straight and tangential to the coordinate directions at the origin of the vector, and the component sizes being in proportion.

Sometimes arrows are drawn for directed line segments.  Those do have a starting and an ending point.  In that case, the shafts of those arrows will follow any curvilinear coordinate system.  Visualizing a directed line segment specified by starting point and displacement vector is rarely done when more than one coordinate is actually changing since then the path along which the displacement is achieved has no obvious candidate: there is no obviously preferred route between starting and ending point any more.