# Distinguish between affine space and vector space

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.

• An affine space can be thought as a vector space without a prefered origin. – C. Falcon Mar 18 '18 at 16:09
• In practice you do computations on the "position vector". But conceptually, points are one thing and vectors are another, especially in physics. – Giuseppe Negro Mar 18 '18 at 16:48

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.

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 .