Can a plane be perpendicular, at the same time, to two other planes if they're not parallel to each other?

Question

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?

Answer 1

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.

Answer 2

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.