For instance in Newtonian physics we treat position of objects, displacements, velocities, forces, momenta, angular velocities etc all as vector quantities (little arrows in space which have a certain direction & magnitude that can be added using the parallelogram law of vector addition & can be multiplied by scalars). But who said that these physical quantities can be modelled using vector algebra? Is this an empirical statement or is there a theoretical reasoning behind it?
For instance why do velocities add linearly in Newtonian Mechanics? If an object is given a velocity of 10m/s North & a velocity of 10m/s East, then according to what I've been taught, the 2 velocities will just add like vectors to give the net velocity of the object but I don't see how this statement is obvious in any way? Is there a deeper reason for why velocity ought to behave like a vector in Euclidean Space?
Moreover as I understand it, the vectors used in Newtonian Mechanics are vectors in 3 dimensional Euclidean Space whereas in relativity theory (where velocities don't add linearly) we use four vectors in an different abstract space, right? If relativity is the more fundamental theory, then why is it that these four vectors in abstract space behave like Euclidean vectors in real space for low velocities? Where's the connection?
When can we use vector algebra to model physical quantities? There are about 5 different definitions of vectors that I've encountered so far. How do we formally define vectors in physics?
Please be as elaborate as possible.