I have a plane $V$ (it's basis) in $\mathbb{R}^3$ and a vector $a$ that belongs to that plane. What should I do when I want to find all planes that intersect $V$ along the line created by $a$? How would I calculate that? I tried visalising some examples:
$$V=\Bigg\{\begin{bmatrix}1\\0\\0\end{bmatrix}, \begin{bmatrix}0\\1\\0\end{bmatrix}\Bigg\}\quad a=\begin{bmatrix}1\\1\\0\end{bmatrix}$$
In this case I think there is an infinite number of planes that intersect $V$ along that line - there is one plane that is orthogonal to $V$ and then there are many, many planes that intersect $V$ under an angle - please tell me if I'm wrong.
I solve your example ( and I hope you can find from this the general method).
In your case the line ''created by $a$ '' is the intersection of the two planes: $$ \begin{cases} x-y=0\\ z=0 \end{cases} $$ All the planes (the bundle of planes) that contains such line are represented by the linear combination of the two planes: $$ \lambda(x-y)+\mu z=0 $$
Note that for $\lambda=0$ we find the plane $V$.