Can two planes intersect in a point?

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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

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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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