Lines and planes in 3D — intersections and distances (IB Math AA HL)

Key ideas

• Line: r = a + λb, where b is the direction. Plane: r · n = a · n, or ax + by + cz = d with normal vector (a, b, c).
• Line meets plane: write x, y and z in terms of λ, substitute into the plane's equation and solve for λ. If λ disappears, the line is parallel to the plane (it lies in it if the equation is still true).
• Angle between a line and a plane: sin θ = |b · n| ÷ (|b||n|). It is a sine, because n is perpendicular to the plane.
• Angle between two planes: the angle between their normals, using cos.
• Two planes meet in a line with direction n₁ × n₂; find one point on it by setting one coordinate to 0.

Worked example

Find the point where the line r = (1, 2, −1) + λ(2, −1, 3) meets the plane x + y + z = 8, and the acute angle between the line and the plane.

1. On the line: x = 1 + 2λ, y = 2 − λ, z = −1 + 3λ.
2. Substitute: (1 + 2λ) + (2 − λ) + (−1 + 3λ) = 8, so 2 + 4λ = 8 and λ = 1.5.
3. Point: (1 + 3, 2 − 1.5, −1 + 4.5) = (4, 0.5, 3.5).
4. b · n = (2, −1, 3) · (1, 1, 1) = 4, |b| = √14 and |n| = √3.
5. sin θ = 4 ÷ √42 = 0.617, so θ = 38.1° (3 s.f.).

Answer: (4, 0.5, 3.5), at an angle of 38.1° to the plane.

Exam-style HL AA questions on this topic

• Planes questions — 47 questions with mark schemes, plus free worked examples
• All HL AA Geometry and Trigonometry questions by topic

Do it on your GDC

A calculator is allowed on AA Paper 2 and Paper 3, not on Paper 1. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:

• Vectors on your calculator

Maths IA ideas that use lines and planes in 3d

• How are letters drawn on a screen? Bézier curves
• Crossing a river with a current: fastest or straightest?
• The hidden angles of a hip roof: 3D trigonometry
• How far does a random walker get? Distance after n steps
• All HL AA IA ideas

Related in Geometry and Trigonometry

• Vector proofs and geometric arguments
• Compound and double-angle formulas
• Trigonometric equations — HL identities in a finite interval

Revise Lines and planes in 3D for the exam

• HL AA revision notes — every unit summarised, with the key results to learn
• HL AA past papers — specimen papers and timed practice papers
• Formulas for Geometry and Trigonometry — on the one-page HL AA formula sheet (free PDF), with the ones not in the IB booklet marked
• HL AA formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

Stretch yourself: original geometry problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S19 · Problem S75 · Problem S18 · Problem S74. All 22 →

Sibling topic in other IB Maths courses