Quotient vector space

Equivalence. $u\sim v$ iff $u-v\in F$, i.e. $u-v=\lambda(1,1,1)$ for some $\lambda\in\mathbb{R}$. So $u$ and $v$ have the same "difference of coordinates": if $u=(x_1,y_1,z_1)$ and $v=(x_2,y_2,z_2)$, then $u\sim v$ iff $x_1-x_2=y_1-y_2=z_1-z_2$.

Classes. Each class can be represented by a vector of the form $(a,b,0)$, where $a=x-z$ and $b=y-z$. The map $E/F\to\mathbb{R}^2$, $[x,y,z]\mapsto(x-z,y-z)$, is an isomorphism.

Dimension. $\dim(E/F)=\dim E-\dim F=3-1=2$.

Conclusion. $E/F\cong\mathbb{R}^2$.