19. Point–Plane Distance
For a plane in defined by and a point , the perpendicular distance from the point to the plane is:
The denominator is the norm of the plane’s normal vector .
The signed distance (drop the absolute value) tells you which side of the plane the point is on:
Example
Plane: . Point: .
The negative sign means the point lies on the opposite side of the plane from the normal direction.
19.1. Generalization
The same formula generalizes to a hyperplane in defined by :
19.2. Connections
- Planes — definition and normal vector
- Projection — same idea: project the displacement vector onto the unit normal
- Hyperplane — -dimensional generalization