A-Level Mathematics: Vectors: Planes, Lines & Intersections

Equation of a plane

General form: r · n = a · n, where n is the normal vector, and a is a point on the plane. Quickly plug in coordinates to test if a point lies on the plane.

Line–plane intersections

Parametric line r = a + λd. Solve simultaneously with plane equation to find λ, then substitute back.

Shortest distance tricks

Use projection formulas: distance from point P to plane is |(P – a) · n| / |n|. For skew lines, distance is |(a₂ – a₁) · (d₁ × d₂)| / |d₁ × d₂|.

Exam habits

• State vectors clearly (column form is neater under time pressure).
• Always normalise your working by underlining final vector answers.
• Watch out for parallel vs coincident cases.