Is there a name/study for this kind of thing?
Edit: I'm not looking for vector addition -- the origins of each arrow should be preserved
The idea is that the vectors are combined so that the end of one vector is the origin of the next vector (all go in the same direction). I'm not that familiar with linear algebra but wanted to learn more about work that's been done on this kind of stuff.
I'm specifically wondering if there's a generalized notion for this kind of thing from abstract algebra.
Yes, there is; it is called addition.
Every vector space by definition has two operations associated with it: Addition of vectors, which is just what you have described, and stretching a vector by a number (i.e. increasing the length by a factor while keeping the same direction).
If you want to learn more about this, you should get familiar with the abstract definitions of groups, fields, and vector spaces (maybe also take a look at inner products). Greetings,
Intergalakti