TITLE: What is a difference space?
QUESTION [5 upvotes]: I've read this sentence on Wikipedia:

The modern approach defines the three-dimensional Euclidean space more
  algebraically, via vector spaces and quadratic forms, namely, as an
  affine space whose difference space is a three-dimensional inner
  product space.

But it doesn't define what a difference space, so I wonder, what is it?

REPLY [4 votes]: It refers to the vector space denoted by $\vec{A}$ in this definition associated with the affine space $A$.
The idea is that while you don't want to assign vectors to each point of affine space, you still want to be able to describe the displacements between the points. The elements of the vector space are used to describe the relative difference between any two points of the affine space.

REPLY [2 votes]: For example: $(5,6,7)$ and $(1,1,1)$ are points in Euclidean $3$-space. Their difference is the vector $\langle4,5,6\rangle$. Any difference of two points in Euclidean $3$-space defines a $3$-vector. The resulting vectors form a vector space on which we can define the usual inner product, resulting in an inner-product space.