Let 
be a vector
space over a field 
, and let 
be a nonempty set. Now define addition
for any vector 
and element 
subject to the conditions:
1. 
.
2. 
.
3. For any 
,
there exists a unique vector 
such that 
.
Here, 
,
. Note that (1) is implied by (2)
and (3). Then 
is an affine space and 
is called the coefficient field.
In an affine space, it is possible to fix a point and coordinate axis such that every point in the space can be represented
as an 
-tuple
of its coordinates. Every ordered pair of points 
and 
in an affine space is then associated with a vector 
.
