I have a set of geometry questions related to objects in $R^3$, and I could really use some assistance in solving them. Here are the questions:
We are studying two geometric objects in $R^3$. The first object, denoted as $P$, is described by the equation $x - y + 2z = 0$, and the second object, $Q$, is described by the equation $x + y = 2$.
(a) What type of geometric representation do $P$ and $Q$ have in $R^3$? Are they points, lines, planes, or volumes?
(b) Are objects $P$ and $Q$ parallel to each other?
(c) Find the equation of the line that passes through the point $[2,-1,-1]^T$ and is parallel to both $P$ and $Q$. Provide the answer in vector parameter form. My Attempt:
For part (a), I believe that $P$ represents a plane in $R^3$ because it's defined by an equation with three variables ($x$, $y$, and $z$). As for $Q$, I think it represents a line in $R^3$ because it has two variables ($x$ and $y$), which suggests it's a 2D object in 3D space.
For part (b), I'm not entirely sure if $P$ and $Q$ are parallel, but I know that to determine this, I need to compare their normal vectors.
For part (c), I'm not sure how to find the equation of the line that passes through $[2,-1,-1]^T$ and is parallel to both $P$ and $Q$. I believe I should find a direction vector that is parallel to both $P$ and $Q$ and then use the point-direction form of a line equation, but I'm uncertain about the exact calculations needed.
Any help and guidance on these questions would be highly appreciated!
I give you some hints, computations are left to you:
a) As they are both defined by an equation in $\mathbb{R}^3$ they both have dimension 2, the former passes through the origin, while the latter does not.
b) Two planes in $\mathbb{R}^3$ intersect in a line if they are not parallel, hence it's sufficient to check if they intersect to answer the question
c) If $P$ and $Q$ are parallel (spoiler, no) there is a whole plane passing through the point you are given which is parallel to $P$ and $Q$, otherwise the line you seek is the one perpendicular to the normal vectors of $P$ and $Q$. Then you just need to find the vector spanning the line and translate it on the point you are given.