Key Idea: This topic compares two lines in 3D: the angle between them, and whether they are parallel, intersecting or skew. It is a standard part of the big 3D vectors question on Paper 1 and Paper 2.
Angle between two lines
- Use only the direction vectors.
- cos θ = |d₁·d₂|/(|d₁||d₂|). The bars give the acute angle.
- d₁·d₂ = 0 means the lines are perpendicular.
Classifying lines: parallel, intersecting, skew
- Directions — d₂ = k·d₁? Then parallel (or the same line).
- Two equations — Use s for one line, t for the other. Solve two rows.
- Third row — Holds: they intersect. Fails: they are skew.
Skew means not parallel and the equations are inconsistent. State both.
Angle between directions (2, 2, 1) and (1, −2, 2)? The dot product is 0, so 90°.
Why use two different letters, s and t? Each line has its own parameter.
Directions are not multiples and the third row fails. What are the lines? Skew.
Lines (1, 2, 1) + s(3, a, 1) and (b, 1, 7) + t(1, 2, −5) meet at right angles. Find a and b. a = 1, b = 3.