Linear Algebra
1.912 min read

Three Variables and Plane Geometry

When we extend to three variables, the geometry changes dimension. A single linear equation a1x1+a2x2+a3x3=ba_1 x_1 + a_2 x_2 + a_3 x_3 = b in R3\mathbb{R}^3 defines a plane rather than a line.

A system of equations in three variables asks: where do multiple planes intersect? Two generic planes in R3\mathbb{R}^3 intersect along a line (one-dimensional). Three generic planes intersect at a single point (zero-dimensional, unique solution).

But exceptional configurations abound in 3D. Three planes might: all share a common line, form a "prism" with no common point, or two might be parallel. The richness of 3D geometry foreshadows the full theory of linear systems.

Formal View

Theorem 1.9 — Plane Intersections
For a system of mm equations in n=3n = 3 unknowns with rank rr, the solution set has dimension 3−r3 - r: \begin{itemize} \item r=3r = 3: unique point (0-dimensional) \item r=2r = 2: line (1-dimensional) \item r=1r = 1: plane (2-dimensional) \item r=0r = 0: all of R3\mathbb{R}^3 (3-dimensional, only if b=0\mathbf{b} = 0) \end{itemize} The system is inconsistent if elimination produces 0=c0 = c with c≠0c \neq 0.

Interactive Visualization

Three planes meeting at a point

Why This Matters

3D systems model three-dimensional physical reality: forces, positions, flows.

  • Structural engineering: three force equilibrium equations at a joint determine the unique force state
  • 3D navigation: three GPS satellite distance equations locate a receiver in space
  • Computer graphics: ray-plane intersection is a 3D linear system solved millions of times per frame

Quiz

Question 1

Two distinct, non-parallel planes in R3\mathbb{R}^3 intersect in:

Common Mistakes

  • Assuming three planes always meet at a point — they can form a "prism" with no common intersection.
  • Forgetting that two planes can be parallel, giving no intersection at all.