AHL 3.18 — Intersections & Angles in 3D

In three–dimensional space, lines and planes can intersect in different ways.
This topic explains how to:

  • find the intersection of a line with a plane,
  • find the intersection of two planes or three planes,
  • calculate the angle between a line and a plane, and the angle between two planes.

Basic forms used for intersections

We usually work with the following equations:

Line (vector / parametric form)
r = a + t d
where a is a point on the line, d is a direction vector and t is a real parameter.

Plane (Cartesian form)
A x + B y + C z = D
where (A, B, C) is the normal vector to the plane.

Intersections are found by substituting the coordinates of one object into the equation of the other and solving the resulting equations.

Intersection of a line with a plane

To find where a line meets a plane:

  1. Write the line in parametric form: x, y, z in terms of t.
  2. Substitute these expressions into the plane’s equation.
  3. Solve for t.
  4. Substitute the value of t back into the line to get the intersection point.

Example 1 — Line with a plane

Line: r = (1, 2, 0) + t(2, −1, 3)
Plane: x + y + z = 6

From the line:
x = 1 + 2t,   y = 2 − t,   z = 3t

Substitute into the plane:
(1 + 2t) + (2 − t) + 3t = 6
3 + 4t = 6 ⇒ 4t = 3 ⇒ t = 0.75

Intersection coordinates:
x = 1 + 2 × 0.75 = 2.5
y = 2 − 0.75 = 1.25
z = 3 × 0.75 = 2.25

So the line meets the plane at (2.5, 1.25, 2.25).

Intersection of two planes

Two planes in 3D can:

  • be parallel (no intersection or coincide),
  • intersect in a line.

To find the line of intersection:

  1. Solve the two plane equations simultaneously for x, y, z.
  2. Express the solution with one parameter (usually set one variable equal to t).
  3. Write the resulting line in vector or parametric form.

Example 2 — Two planes intersecting in a line

Plane 1: x + 2y + z = 6
Plane 2: 2x − y + z = 3

Subtract Plane 1 from Plane 2:
(2x − y + z) − (x + 2y + z) = 3 − 6
x − 3y = −3 ⇒ x = 3y − 3

Substitute into Plane 1:
(3y − 3) + 2y + z = 6
5y − 3 + z = 6 ⇒ z = 9 − 5y

Let y = t, then
x = 3t − 3,   z = 9 − 5t

So the line of intersection is:
r = (−3, 0, 9) + t(3, 1, −5).

Intersection of three planes

Three planes can:

  • meet at a single point (unique solution),
  • intersect along a line (infinitely many solutions),
  • have no common intersection (parallel or inconsistent system).

This is analysed by treating the three equations as a system of linear equations in x, y, z.
The number and type of solutions correspond to the geometry of the planes.

Angle between a line and a plane

Let a line have direction vector d and a plane have normal vector n.
The angle φ between the line and the plane is related to the angle θ between d and n by:

θ = angle between d and n, given by
cos θ = (d · n) ÷ (|d| × |n|).

The angle between the line and the plane is
φ = 90° − θ.

Example 3 — Angle between a line and a plane

Plane: x + 2y + 2z = 7 has normal n = (1, 2, 2).
Line: direction vector d = (2, 1, −1).

d · n = 2×1 + 1×2 + (−1)×2 = 2 + 2 − 2 = 2
|d| = √(22 + 12 + (−1)2) = √6
|n| = √(12 + 22 + 22) = √9 = 3

cos θ = 2 ÷ (√6 × 3) = 2 ÷ (3√6).
θ is the angle between d and n.
The angle between the line and the plane is φ = 90° − θ.

Angle between two planes

The angle between two planes is defined as the angle between their normal vectors.

If planes have normals n1 and n2, then
cos θ = (n1 · n2) ÷ (|n1| × |n2|).

Example 4 — Angle between two planes

Plane 1: x + y + z = 0 → normal n1 = (1, 1, 1)
Plane 2: 2x − y + 2z = 5 → normal n2 = (2, −1, 2)

n1 · n2 = 1×2 + 1×(−1) + 1×2 = 3
|n1| = √(1 + 1 + 1) = √3
|n2| = √(4 + 1 + 4) = √9 = 3

cos θ = 3 ÷ (√3 × 3) = 1 ÷ √3
So θ is the angle between the two planes.

🌍 Real-world connection

  • Aviation: flight paths (lines) intersect altitude levels or approach surfaces (planes).
  • Architecture: roof faces and wall faces are modelled as planes; their intersections define edges.
  • Computer graphics: ray–tracing algorithms repeatedly compute intersections of rays (lines) with surfaces (planes).
  • Robotics: to avoid collisions, robot arms are modelled with line segments intersecting safety planes.

🔍 TOK perspective

  • Symbolic equations often make it easier to reason about 3D intersections than drawings do. Does abstraction increase or decrease our understanding?
  • The same configuration can be represented as equations, vectors, or diagrams. How does the choice of representation shape what we notice or ignore?
  • Intersections in higher dimensions cannot be visualised directly. To what extent is our mathematical knowledge independent of our visual intuition?

📊 IA spotlight

  • Analyse intersections of real flight routes or railway lines using publicly available coordinate data.
  • Investigate how changing coefficients in plane equations affects the type of intersection (point, line, or none).

🌐 EE focus

  • Study systems of linear equations from a geometric viewpoint — lines and planes in higher dimensions.
  • Explore applications of line–plane intersections in computer vision or 3D mapping as a bridge between mathematics and technology.

❤️ CAS link

  • Create a workshop for younger students where they build 3D models showing how lines and planes intersect.
  • Help a school club design stage sets or structures using simple 3D sketches and intersection calculations.

🧠 Examiner tip

  • Always write out the parametric form of a line clearly before substituting into a plane.
  • When finding line–plane intersections, check whether the direction vector is perpendicular to the normal: if d · n = 0 the line is parallel to the plane.
  • For angles, state clearly whether you are finding the angle between normals (planes) or between a direction vector and a normal (line–plane).

📱 GDC use

  • Use the simultaneous equation solver to quickly solve systems for intersections of two or three planes.
  • Use built-in dot-product and norm functions to compute angles accurately.
  • Graph parametric lines and implicit planes (if supported) to visualise intersection behaviour.

📝 Paper 2 strategy

  • Write each step when solving simultaneous equations; partial working can earn method marks even if arithmetic slips.
  • In “interpret the solution” questions, link algebraic results to geometry: point, line, or no intersection.
  • Clearly label final answers as “intersection point”, “line of intersection” or “angle between planes”.