Improper pencil

Property — Improper pencil

All the planes parallel to $\pi\colon ax+by+cz+d=0$ have equation $ax+by+cz+k=0, \qquad k\in\mathbb{R}:$ the normal vector $(a,b,c)$ is kept fixed and only the constant term varies.

Parallel planes share the same normal vector: this is why only the constant term $k$ changes.

Links

Topics: Analytic geometry in space
Concepts: Pencil of planes · Plane
Methods: Improper pencil of planes
Skills: Analytic geometry · Using formulae