Well, my intuition says they are always parallel, but the text book says they are not. I am quite shocked to be honest.
Does anyone know a case they are not parallel and perpendicular to a third plane at the same time?
Also, two planes that are parallel to a line must also parallel to each other in my opinion, and the text book says they are not always parallel.
Does anyone know why?
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In $\mathbb R^3$, a simple counterexample for the first assertion is the three standard coordinate planes, which are pairwise perpendicular.
For the second assertion, consider the $z$-axis. The planes $x=1$ and $y=1$ are both parallel to the $z$-axis, but are perpendicular to each other. Being parallel to a line still leaves a degree of freedom in choosing the direction of the plane.
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Take any two planes $\pi_1$ and $\pi_2$ that are not parallel to each other. The two planes $\pi_1$ and $\pi_2$ intersect in a line. Call that line $\ell$.
Then any plane perpendicular to $\ell$ is also perpendicular both to $\pi_1$ and to $\pi_2.$
Also, any line parallel to $\ell$ (including $\ell$ itself) is parallel both to $\pi_1$ and to $\pi_2.$
No. Of course not. Place a hardback book standing up on your desktop and open the two hard covers to some arbitrary angle. The planes of the covers are each perpendicular to the desk top, but not parallel to each other (or necessarily perpendicular to each other).