Distinguish between affine space and vector space

Question

Why it's important to distinguish the concepts of affine space and vector space ? Why can't be condensed in unique concept ? I'm a physics students and these seems too abstract concepts to me, since in physics we simply consider points as vectors and vectors as points.

Answer 1

There are physical concepts that can be better represented by the structure of an affine space, as the usual 3D space of classical physics, but other are better represented in a vector space structure, as the forces acting on a point , that are ususally vectors in $\mathbb{R}^3$.

There are also more ''exotic'' situations, as the vector Hilbert space of the states in Quantun Mechanics .

Answer 2

An affine space is not a vector space but it is a shifted vector space.

Let us look at the xy- plane which is a two dimensional vector space.

A straight line which goes through the origin is a one dimensional subspace and it a vector space.

What about a straight line which does not go through the origin?

It is not s subspace because it does not contain the $0$ vector.

But you may shift it to contain the origin and the shifted version is a vector space.

We call it an affine space because it is a shifted vector space.

For example $y= 3x+10$ is an affine space because it is a shifted version of $y=3x$ which is a vector space.