# Can two planes intersect in a point?

Is it true that two planes may intersect in a point ?

or

If they intersect then, they always make a straight line ?

I have some doubt; please explain.

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## 1 Answer

In $\Bbb R^3$ two distinct planes either intersect in a line or are parallel, in which case they have empty intersection; they cannot intersect in a single point. In $\Bbb R^n$ for $n>3$, however, two planes can intersect in a point. In $\Bbb R^4$, for instance, let $$P_1=\big\{\langle x,y,0,0\rangle:x,y\in\Bbb R\big\}$$ and $$P_2=\big\{\langle 0,0,x,y\rangle:x,y\in\Bbb R\big\}\;;$$ $P_1$ and $P_2$ are $2$-dimensional subspaces of $\Bbb R^4$, so they are planes, and their intersection $$P_1\cap P_2=\big\{\langle 0,0,0,0\rangle\big\}$$ consists of a single point, the origin in $\Bbb R^4$. Similar examples can easily be constructed in any $\Bbb R^n$ with $n>3$.

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Seems correct. Still, how do we demonstrate that two planes in $\mathbb{R}^3$ cannot intersect in a single point. –  Teddy Oct 23 '12 at 8:40
@Teddy WLOG remove one zero from P1 and P2. Now there's no way to not get $x=y$, unless they describe the same plane. –  Mark Hurd Oct 23 '12 at 8:45
@MarkHurd I think I understand your intention. However, this reasoning only demonstrates that planes of this particular type intersect in a line. What about other planes. What about planes which are not subspaces of $\mathbb{R}^3$ (but rather affine spaces) ? –  Teddy Oct 23 '12 at 9:38
@Teddy: Simultaneously translate them by the additive inverse of a point of intersection, thereby bringing that point to the origin. Now you’re dealing with subspaces, and their intersection can be translated back to give the intersection of the original planes. –  Brian M. Scott Oct 23 '12 at 9:42