18. Planes
A plane in is the set of points satisfying a single linear equation:
The vector is the normal vector — perpendicular to every vector lying in the plane.
18.1. Point–normal form
If is a known point on the plane and is the normal, then any other point lies on the plane iff is perpendicular to :
Expanding componentwise:
Example
Find the plane equation given and a point on the plane .
Simplify:
18.2. Reading the normal off the equation
In the form , the coefficients are exactly the components of the normal vector:
Example
18.3. See also
- Point–Plane Distance — formula and example
- Distance Between Planes — parallel and non-parallel cases
- Hyperplane — -dimensional generalization
- Cross Product — building a normal from two in-plane vectors