Key Idea: This topic finds where lines and planes cross, and the angles between them. It is a standard step in the long vectors question on Paper 1 and Paper 2.
| After you substitute the line into the plane | Meaning |
|---|---|
| One value of λ | One point: put λ back into the line |
| A false statement, e.g. 2 = 5 | Parallel, no meeting |
| A true statement, e.g. 3 = 3 | The line lies in the plane |
Planes meeting & angles
- Line of two planes: direction n₁ × n₂; get a point by setting z = 0.
- Plane to plane: cos θ = |n₁·n₂|/(|n₁||n₂|).
- Line to plane: sin θ = |d·n|/(|d||n|).
Two planes use cosine. A line and a plane use sine.
Where does r = (3, −1, 4) + λ(1, 2, −1) meet 2x − y + 3z = 7? λ = 4, so P = (7, 7, 0).
Where do you put λ back, the line or the plane? The line.
Angle between 2x + y − 2z = 4 and x + 2y + 2z = 1? n₁·n₂ = 0, so 90.0°.
Why sine for a line and a plane? The normal is 90° from the surface.