Three planes and a common point

Statement

Show that, given three planes that do not contain the same line, there is in general a unique common point. What happens if the three planes belong to the same pencil?

Solution

Three planes give a linear system of three equations in the three unknowns $x,y,z$. If the three normal vectors are linearly independent (the planes are not “aligned” around a line), the system is determinate and has a single solution: the three planes meet at a single point, like three walls converging along an edge.

If instead the three planes belong to the same proper pencil, then they all contain the axis line $r$: every point of $r$ satisfies all three equations. The system is indeterminate and the common points are infinitely many (the whole line $r$), not just one. If they belong to an improper pencil they are parallel and have no point in common.

Links

Topics: Analytic geometry in space
Concepts: Pencil of planes · Plane
Skills: Proving · Analytic geometry · Reasoning by cases
Exercise types: Proof