[ajkjk]

One finds in regular vector algebra that "position vectors" and "displacement vectors" are sort of two distinct types of objects, and that it is never physically valid to add two position vectors together unless you create an affine combination like (a+b)/2. A position vector 'a' is really 'O + a', so [(O+a) + (O + b)]/2 = O + (a+b)/2, another position... but a+b on its own would really be (O +a) + (O + b) = 2O + a + b, which is not geometrically meaningful. So positions and displacements might both be elements of R^2, mathematically speaking, but there is something physically different about them, which physical applications/geometry forces you to contend with. I think it is something like a historical accident that there's not a great notation for expressing this in normal mathematics (or at least, I'm not aware of one!).

[gugagore]

https://math.ucr.edu/home/baez/torsors.html

The distinction is whether zero is meaningful independent of a choice of origin. Zero displacement is meaningful. Zero position is arbitrary.

Are you thinking of displacement as an operation? Because it is just as well a vector. I don't see the connection to section I highlighted from the article.

[ajkjk]

I think of displacement vectors and position vectors as having different "types". Position vectors are geometric objects whose meaning comes from whatever physical system you're studying (this way of thinking is generally useful even if you're not doing physics; for instance if you're talking about a manifold you would think of that as the physical system independent of the coordinates you put on it).

In the same way I think of "vectors as operators" (rotations/scaling) as a displacement vector / torsor, but of a different type than their sense as translations. As far as I can tell, the geometric product between two displacement vectors is not so meaningful, whereas the geometric product between two "operator" vectors is (because it composes them as operators, in some sense). But in practice you're often rapidly switching between these representations so it's hard to tell which object you're actually talking about. For this reason I find it useful to distinguish their types explicitly.