I need to find the equation of a plane that passes through a point and is perpendicular to two other planes (equations were given). To my understanding, if two planes are not parallel to each other, there is no plane or line that is, at the same time, perpendicular to those planes individually. So I need to interpret this as a plane that is perpendicular to the intersection of these two?
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2Think about the coordinate planes. The $xy$ plane is perpendicular to the other two coordinate planes, yet they are not parallel. – Cheerful Parsnip Feb 11 '22 at 04:25
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In 3D the line perpendicular to both planes is impossible, but the perpendicular plane always exists (it is perpendicular to the intersection line of the first two planes). – user Feb 11 '22 at 09:45
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Use the cross product of the normals of the two given planes, and the given point that the third plane passes through, to form the dot-product vector equation of the third plane.
ryang
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The cross product works in 3 dimensions, but in higher dimensions, it isn't available. – Brian Borchers Feb 11 '22 at 04:24
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1@BrianBorchers But given $n-1$ linearly independent vectors in $n$-dimensional space we can find the vector orthogonal to all of them using essentially the same technique (determinant) as for the cross product. – user Feb 11 '22 at 09:52
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It is possible for a plane in three-dimensional space to be perpendicular to two other non-parallel planes, and indeed even for three planes to all be perpendicular to each other: a simple example, as noted by Cheerful Parsnip in the comments above, would be the three coordinate planes (defined by the equations $x = 0$, $y = 0$ and $z = 0$ respectively).
In fact, any plane that is perpendicular to the line of intersection of two planes is also perpendicular to each of those two planes. So, while your reasoning in your question is incorrect, the method you suggest in your last sentence will indeed yield the correct solution.
Ilmari Karonen
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