Describe the collection of $L$'s right cosets

Question

Given the group $G = (\mathbb{R}^{2}, +)$, I know that if $L$ is a line going through the origin, so $L$ is a sub-group of $G$. My question is: what is the simplest way to describe the collection of $L$'s right cosets?

Answer 1

Since additivity here is commutative, the left cosets are the same as the right cosets, so the right cosets here are the lines parallel to $L$, each line having as representative a point that lies on the line going through the origin perpendicular to $L$.