How to interpret geometrically this space? Good Evening. I'm trying to do this exercise, but I have many doubts. Let be $\mathbb{R}^3$ and $W=\{(x,y,z)\in\mathbb{R}^3:x+y+z=0\}$ a subspace of $\mathbb{R}^3$. Find the quotient vector space $\mathbb{R}^3/W$
My work:
I need to find the equivalence class of $'u'$ a generic or vector belonging to the quotient. So I thought $u=(a,b,c),v=(d,e,f)$ both in $\mathbb{R}^3$ then $uRv$ iff $u-v\in W$ So I cleared the variable $x$ of $W$ and write the plane as $W=\{(-y-z,y,z):y,z\in \mathbb{R}\}$ and I got it $[u]=\{(y+z+a,b-y,c-z):y,z\in \mathbb{R}\}$, where $[u]$ represents the equivalence class of $u$, then $\mathbb{R}^3/W=\{[u]:u\in \mathbb{R}^3\}$ I do not understand the partition that generates this set on $\mathbb{R}^3$, I do not know what geometrical interpretation also provides. How I find the equivalence class vector (1,2,3)? Help me please...Thanks!
 A: $W = \{(x,y,z) \in \mathbb{R}^3 : x + y + z = 0 \}$ is a plane passing through $(0,0,0)$, normal to the vector $(1,1,1)$. 
For $u = (u_1, u_2, u_3)$, geometrically $u + W$ is the plane $W$ translated by $u$. Equivalently, it is the plane normal to $(1,1,1)$, passing through the point $(u_1,u_2,u_3)$.
Algebraically,
$$
[u] = \{(x,y,z) \in \mathbb{R}^3 : x + y + z = u_1 + u_2 + u_3 \}.
$$
For $u = (1,2,3)$, this set of points is
$$
[(1,2,3)] = \{(x,y,z) \in \mathbb{R}^3 : x + y + z = 6\},
$$
which has the geometric interpretation as I explained above. 
A: Here is your plane $W = \{(x, y, z): x + y + z = 0\}$ in green and vector $v = (1, 2, 3)$ in blue.

Now we add the plane $W' = \{(x, y, z): x + y + z = 10\}$ in dark blue (shown as a triangular grid so you can see through it).

And a secondary shot, so you can see the two really are parallel, with the terminal point of $v$ lying on $W'$.

Now we'll add some vectors $v_i'$ whose terminal points are also on $W'$, in red. Note that $v - v_i' \in W$, and are thus in the equivalence class $[(1, 2, 3)]$, since the terminal points of both are in $W'$. They really do differ by a vector in $W$, and we can see: the line segment joining them is in the plane $W'$, with $W'$ parallel to $W$.

And, for good measure, another perspective.

What do other equivalence classes look like? All the vectors whose terminal points are in some plane parallel to $W$, with these planes having varying distance from $W$. Here are several of these planes, in various colors (with the original plane in green still):

In fact, for each distance $d \in \Bbb R$, there is a unique plane $W_d$ parallel to $W$ that is distance $d$ from $W$: it is the plane $W_d = \{(x, y, z): x + y + z = \frac{d}{\sqrt{3}}\}$. The set of all vectors whose terminal points lie in a plane $W_d$ constitute an equivalence class $[\frac{1}{\sqrt{3}}(d,d,d)]$ in $\Bbb R^3 / W$. 
