This is regarding the following question from Greub's Linear Algebra:
Let $V$ be a plane and $U$ a straight line through the origin. What is the geometrical meaning of the elements of the quotient space $V/U$? Give a geometrical interpretation of the fact that $x \sim x'$ and $y \sim y' \implies x+y \sim x'+y'$ ?
Approach for first part: $U$ contains all vectors of the form $\lambda v$ for some $v \in V$. So if $x$ lies in an equivalence class w.r.t. $U$, $x \sim y$ for some $y \in V \implies x-y=\lambda v$, or $x = y+\lambda v$ for some scalar $\lambda$ and some $y \in V$. Therefore, that equivalence class is a line parallel to $U$.
I'm not sure about the second part though. The only geometrical interpretation of the sum of 2 vectors ($x+y$) that I know of is that it represents the point opposite to the origin in the parallelogram formed by two lines joining the origin to $x$ and $y$. Is there an interpretation independent of the origin's location? I just can't think of a simple interpretation of the given result and I'm not even sure whether it has anything to do with transformation of the coordinate axes.
Any help with the second part would be appreciated.