Intersections, perpendiculars and distances with vectors

Last time

Before we start: last time, in Vector equations of lines, skew lines and the angle between lines, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Geometry with arrows

Every geometry question in three dimensions turns into algebra once you write points as vectors. Where two lines cross is an equation. The foot of the shortest path from a point to a line is a dot product equal to zero. Learn those two moves and you can answer most of Question 9.

Finding the crossing point

Write both lines with different parameters. Equate the x components, the y components and the z components. Solve any two equations for lambda and mu, then check the third. Substitute lambda back into its line to get the position vector. Quote it as a vector, or as coordinates if the question asks.

A past-paper example

Oct Nov 2025, Paper 33, question 9. Line l one is 3, 1, minus 6 plus lambda times 2, 1, 4. Line l two is minus 1, 3, minus 6 plus mu times 0, 1, 2. The x components give 3 plus 2 lambda equals minus 1, so lambda is minus 2. The y components give 1 plus lambda equals 3 plus mu, so mu is minus 4. Check z. Minus 6 plus 4 lambda is minus 14. Minus 6 plus 2 mu is also minus 14. It holds. The point of intersection is minus i, minus j, minus 14 k.