Geometric interpretation of the quotient vector space Good evening, 
I'm doing this exercise but I have some doubts on the geometric interpretation of this vector space.
Let be $\mathbb{R}^3$ and the subspace of $\mathbb{R}^3$ $W=\{(x,y,z)\in\mathbb{R}^3:2x-y+2z=0\}$. Find the vector quotient space $\mathbb{R}^3/W$, the class of "u" a generic vector in the quotient and provide a geometric interpretation of this vector space.
My work:
I started to clear the variable z of the subspace W, and I got $z=\frac{-2x+y}{2}$, now $W=\{(x,y,\frac{-2x+y}{2})\in\mathbb{R}^3\}$ then take a vector $u=(a,b,c)$ and $v=(d,e,f)$ both in $\mathbb{R}^3$, after $vRu$ if and only if $v-u \in W$ then $v-u=(d-a,e-b,f-c)=(x,y,\frac{-2x+y}{2})$ and my question is the class of $u$ is $[u]=\{(x+a,y+b,\frac{-2x+y}{2}+c)|x,y \in \mathbb{R}\}$ ? I have doubts and i have no idea of the geometric interpretation. Thanks!
 A: Let $V = \mathbb R^3$. We know that $W$ is a plane.
Since $V$ is finite dimensional, $$\operatorname{dim}(V/W) = \operatorname{dim}V - \operatorname{dim}W = 3 - 2 = 1.$$ This tells us that the quotient space should be a line.
To get some further geometric intuition I think it's enough to look at the plane given by the equation $z=0$, that is from now on $W = \{ (x,y,0) : x,y \in \mathbb R \}$. 
Now $u,v \in \mathbb{R}^3/W $ iff $u-v \in W$. This tells us that two vectors are considered equivalent iff they have the same 3rd coordinate, that is they are in the same plane parallel to $W$.
Imagine that an element of the quotient space is made by compressing an equivalence class into a single point. In this case an element of $V/W$ is made by compressing a plane parallel to $W$ into a single point. This results in a single line which I personally imagine to be the $z$ axis.
Finally for a bad analogy, imagine $\mathbb {R}^3$ as a cylinder symmetric to the $z$ axis with an infinite height and radius. Then forming $\mathbb {R}^3 / W$ is akin to shrinking the radius of this cylinder to $0$, thus getting the $z$ axis. 
