How to find the position of a point relative to a plane?

Question

Mathematics

The task was to find the position of the point relative to the plane. Possible positions: on the plane, above the plane and below the plane.
Everyone is familiar with the equation of the plane.
Ax + By + Cz + D = 0
And everything seems to be simple, we remember from the course of school geometry, if we take the coordinates (x, y, z) of the desired point and substitute it into the equation, we can understand where the point is. If =0 means on the plane, <0 under the plane, >0 above.
Okay, let's consider a plane 2x + 3y + 4z + 5 = 0and a point K(0, 0, 0). We get, 5 > 0 - the point lies above the plane. Great, now multiply the left and right sides of the plane equation by -1. We obtain an equivalent equation -2x - 3y - 4z - 5 = 0for our plane. Let's check the point K(0, 0, 0) again. We get -5 < 0 - the point lies below the plane.
I know I'm making a stupid mistake somewhere, but I can't figure out where. Tell me please.

Answer 1

Pleasant questions, even if you're fooling around, they gave me pleasure.
Multiplying by -1 is the same as reflecting and flipping. When reflected, the "right trio of vectors" (google the phrase) becomes left, the flip does not change them.
Multiplying by 5 will correspond to space compression. However, this is meaningful only when both new and old coordinates exist at the same time, for example, having a terrain and its plan, it is easy to understand and apply the concept of "scale".

Answer 2

If "above-below" is the position along the Y axis, then you need to find a point in the K' plane, with the x and z coordinates equal to the corresponding coordinates of the point K being checked and compare the K' and K coordinates.
For your case, you need to solve the equation
2 * 0 + 3 *y+4*0+5 = 0, y _K' = -1.25, y _K' < y _K , the point lies above the plane.
-2*0-3*y-4*0-5 = 0, y _K' = -1.25, y _K' < y _K , the point lies above the plane.

Answer 3

You have xK and yK coordinates of point K, if you change z you get a "vertical" line that "pierces" the plane (unless it is parallel to it) at some point with coordinate z'. You can find z' from the equation of the plane by substituting xK, yK into it, and then compare which of the coordinates: z' or zK above / below.