Injectivity of sums on two distinct rays
= Injectivity of sums on two distinct rays
{title2=$|P+Q|=|P||Q|$}
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.