I am aware of the geometric interpretations of dot and cross products of 2D vectors.
Considering a rational numbers $v1=(x,y)$ and $v2=(u,v$) as 2D vectors;
Are there any known geometric interpretations of the sum of two rational numbers and their difference?
The sum $v1+v2=(x*v+u*y, v*y)$.
P.S. I am only interested in 2D interpretation.