Quotient space: $\mathbb{R}^3$ by a line.

Question

I am trying to understand, geometrically, a plot of the quotient of $\mathbb{R}^3$ quotiented by some line $L$, through the origin. The Wikipedia article states;

Similarly, the quotient space for $\mathbb{R}^3$ by a line through the origin can again be represented as the set of all co-parallel lines, or alternatively be represented as the vector space consisting of a plane which only intersects the line at the origin.

$L$ is a one-dimensional subspace, so this quotient space must surely be two-dimensional. However, a set of lines, as in the case of quotienting $\mathbb{R}^2$ by a line $L$, is one-dimensional. I'm having trouble both understanding this and visualizing the result.

As another matter, I don't want to just take for granted that the equivalence classes are sets of "co-parallel" lines, but my attempt to prove this by picking an arbitrary point $(x,y,z)$ in $\mathbb{R}^3$ and considering the set of $(a,b,c)$ such that $(x,y,z) - (a,b,c) \in L$ didn't produce a workable result (in fact, it produced a plane, rather than a line).

Answer 1

Consider a line $L$ passing throw origin, as a vector subspace of $\mathbb{R}^3$,
so the quotient space will be the space of equivalent relations that is geometrically a plan perpendicular to $L$, which is a $2$-dimensional vector subspace in $\mathbb{R}^3$.

Answer 2

We can assume that the line $L$ goes in the $x$-direction, so it is generated by $(1,0,0)$. This does not change the outcome, since this can be achieved by a simple linear transformation.

Then you want to know what the space $\mathbb R^3$ looks like, when you impose the equivalence relation $v \sim w$ iff $v-w \in L$. You can also phrase this relation like: $v \sim w$ iff $v = w + (a, 0, 0)$ for some $a \in \mathbb R$. This means that the coset of $ w $ (the set of elements that are equivalent to $w$) is a line, parallel to $L$, through $w$. This is what wikipedia means when they talk about co-parallel lines: you partition the space into lines that are parallel to $ L $.

For each coset we can pick a unique representative $ w = (0, a, b) $ with $ (a, b) \in \mathbb R^2 $ (this unique representative is just the intersection of the co-parallel line with the plane $x=0$). The quotient space is the space of the cosets, is the space of the representatives, is $ \mathbb R^2 $.

Does this make sense?