Key Idea: A line in vector form is r = a + λd: a start point plus a direction. This topic builds lines and finds points on them, including moving objects. It is on Paper 1 and Paper 2.
Vector equation of a line
- a is any point the line passes through; d is any multiple of the direction.
- Two points: d = (second point) − (first point).
- Parametric form: each row as its own equation, all sharing λ.
| Job | How |
|---|---|
| Is a point on the line? | One λ must fit every row |
| Point for a given λ | Substitute the number |
| Where it crosses an axis | Other coordinates = 0, solve for λ |
| Moving object | a = start, λ = time, d = velocity, speed = |d| |
Everything on a line comes down to one value of λ.
Direction through P(0, 3, −1) and Q(2, 3, 5)? Q − P = (2, 0, 6).
Speed of a boat with r = (−2, 5) + t(6, 8)? |(6, 8)| = 10 km/h.
Its position after 2 hours? (10, 21).
Why is there more than one correct line equation? Any point and any multiple of d work.