AHL 3.11 (HL)—Vector equations of lines
- Syllabus
- First assessment 2021
- Objective
- —
- Level
- HL
A vector line is a point plus every multiple of a direction
HL onlyA line through point a with direction vector d is r=a+td, where t is a real parameter. Changing t moves along the line without changing its direction.
Use two points P and Q to form d=Q−P. To test whether X lies on the line, solve the component equations for one common t; inconsistent values mean X is not on it.
Through (1,0,2) and (3,4,2), r=(1,0,2)+t(2,4,0). The point (2,2,2) has t=1/2 in every component, so it lies on the line.
Matching one coordinate is not enough. Check the same parameter across all components and distinguish a line from a finite segment.