I keep hearing different answers for what the intersection of two planes is. I believe it is a line, but it can also be a plane IF the two planes are not distinct. However other sources are saying that the intersection of two planes can also be a point or an empty set. So what is the intersection of two distinct, nonparallel planes? A point, line, and/or a empty set?
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$\begingroup$$\newcommand{\Reals}{\mathbf{R}}$For definiteness, I'll assume you're asking about planes in Euclidean space, either $\Reals^{3}$, or $\Reals^{n}$ with $n \geq 4$.
The intersection of two planes in $\Reals^{3}$ can be:
- Empty (if the planes are parallel and distinct);
- A line (the "generic" case of non-parallel planes); or
- A plane (if the planes coincide).
The tools needed for a proof are normally developed in a first linear algebra course. The key points are that non-parallel planes in $\Reals^{3}$ intersect; the intersection is an "affine subspace" (a translate of a vector subspace); and if $k \leq 2$ denotes the dimension of a non-empty intersection, then the planes span an affine subspace of dimension $4 - k \leq 3 = \dim(\Reals^{3})$. That's why the intersection of two planes in $\Reals^{3}$ cannot be a point ($k = 0$).
Any of the preceding can happen in $\Reals^{n}$ with $n \geq 4$, since $\Reals^{3}$ be be embedded as an affine subspace. But now there are additional possibilities:
- The planes $$ P_{1} = \{(x_{1}, x_{2}, 0, 0) : x_{1}, x_{2} \text{ real}\},\qquad P_{2} = \{(0, 0, x_{3}, x_{4}): x_{3}, x_{4} \text{ real}\} $$ intersect at the origin, and nowhere else.
- The planes $P_{1}$ and $$ P_{3} = \{(0, x_{2}, 1, x_{4}): x_{2}, x_{4} \text{ real}\} $$ are not parallel (in the sense that neither is a translate of the other), but they do not intersect.
The planes $P_{1}$ and $P_{3}$ are "partially parallel" in the sense that there exist parallel lines $\ell_{1} \subset P_{1}$ and $\ell_{3} \subset P_{3}$. This turns out to be true for every pair of disjoint planes in $\Reals^{4}$.
In $\Reals^{5}$, there exist "totally skew" planes, such as $$ P_{4} = \{(x_{1}, x_{2}, 0, 0, 0)\},\qquad P_{5} = (0, 0, 1, x_{4}, x_{5})\}. $$
(The terms "partially parallel" and "totally skew" are not standard as far as I know.)
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