I've been working on a math problem involving two geometric objects in $\mathbb R^3$. The first object, which we'll call $P$, is described by the equation $x - y + 2z = 0$, and the second object, $Q$, is described by the equation $x + y = 2$.
I've successfully found the eigenvalues for these objects, and I'm wondering what the next step should be in my analysis. Specifically, I have a few questions:
(a) Do $P$ and $Q$ represent points, lines, planes, or volumes in $\mathbb R^3$?
(b) Are $P$ and $Q$ parallel to each other?
(c) How can I find the line that passes through the point $[2, -1, -1]^T$ and is parallel to both $P$ and $Q$? Please provide the answer in vector parameter form.
I would greatly appreciate any guidance or insights on how to proceed with these questions. Thank you in advance!