Euclidean ray ($\{tv:t\ge 0\}$)

A Euclidean ray from the origin with direction $v\ne 0$ is the geometric half-line $\{tv:t\ge 0\}$. Points in the strictly positive quadrant lie on rays with positive slope.

If finite point sets $P,Q$ lie on two distinct Euclidean rays in a plane, the map $(p,q)\mapsto p+q$ is injective because the two direction vectors are linearly independent. Thus $|P+Q|=|P||Q|$. If all points are nonzero and the rays have positive slope, every sum lies in the open planar sector between those rays.

The open planar sector between two distinct Euclidean rays of positive slope consists of nonzero points whose directions lie strictly between their directions. Sectors between consecutive members of an ordered collection of Euclidean rays are disjoint.