AHL 3.17 — Vector Equations of a Plane

A plane in three–dimensional space is a flat surface that extends infinitely in two directions.
To describe a plane, we need:

  • a point on the plane, and
  • either two non–parallel directions in the plane, or one normal (perpendicular) direction to the plane.

📌 1. Vector Equation Using Two Direction Vectors

One way to describe a plane is:

r = a + λb + μc

Meaning of each symbol:

  • a: position vector of a known point on the plane.
  • b and c: two non–parallel vectors lying in the plane.
  • λ, μ: real numbers (parameters) that can take any value.

Conceptually: start at the point represented by a, then move any combination of
b and c. All such points fill the entire plane.

Example 1 — Plane through a point with two directions

Point A(1, 0, 2), direction vectors b = (2, 1, 0) and c = (−1, 0, 3).
A vector equation is:

r = (1, 0, 2) + λ(2, 1, 0) + μ(−1, 0, 3)

Any choice of λ and μ gives a point on the plane containing A, b and c.

📌 2. Vector Equation Using a Normal Vector

A second description of a plane uses a normal vector n (perpendicular to the plane):

r · n = a · n

Here:

  • n is a vector perpendicular to the plane.
  • a is the position vector of a fixed point on the plane.
  • r is the position vector of any general point (x, y, z) on the plane.

Meaning: the component of r in the direction of n is the same as the component of a in the direction of n.
This is exactly what it means for r to lie in the same plane as a with normal n.

📌 3. Cartesian Equation of a Plane

Writing r = (x, y, z), a = (x0, y0, z0) and n = (a, b, c),
the normal form becomes:

(x, y, z) · (a, b, c) = (x0, y0, z0) · (a, b, c)

Which simplifies to the familiar Cartesian equation:

a x + b y + c z = d,

where d = a x0 + b y0 + c z0.

Here (a, b, c) is the normal vector to the plane, and every point (x, y, z) satisfying the equation lies in that plane.

Example 2 — Cartesian equation from point and normal

A plane passes through P(2, −1, 3) and has normal vector n = (1, 2, −1).

Using a x + b y + c z = d with (a, b, c) = (1, 2, −1):

Substitute point P into the left side:
1×2 + 2×(−1) + (−1)×3 = 2 − 2 − 3 = −3

So d = −3 and the plane equation is:
x + 2y − z = −3.

🌍 Real-World Connection

  • Computer–aided design (CAD) and 3D modelling software use plane equations to define walls, floors, and other flat surfaces.
  • In navigation and aviation, planes can represent constant–altitude levels or boundaries between regions of airspace.

🔍 TOK Perspective

  • We can represent the same plane with many different equations or vector forms. What does this say about the relationship between mathematical objects and their representations?
  • When are different forms (vector, normal, Cartesian) more useful, and how does choice of representation shape our understanding of a problem?