In an [affine space](http://en.wikipedia.org/wiki/Affine_space) $A$, the displacement (difference) between two points is a vector, and one can add a vector to a point, but not two points. However these can be replaced by a ternary operation in terms of points alone: the parallelogram rule $\nearrow : A \times A \times A \to A,\\,\nearrow(p, a, b) = p+(a-b)$.

You can even add scalar multiplication of the difference into the bargain.

Why would you want to do this? Well affine spaces are more primitive than a vector space -- yet we use a vector space in defining them. To me the more natural approach is to define them without it, and watch the vector space (of displacements) drop out.