Stand at a point, then walk in a fixed direction: Imagine standing at a fixed point a and walking in a straight, fixed direction d.
Every point on the line is a + (some number) × d. That number is the parameter λ (lambda): λ = 0 puts you at a, λ = 1 takes one step along d, λ = −1 steps backwards, and fractions land you in between.
So the line is the set of all positions r = a + λd as λ runs over every real number.
Two important freedoms: Any point on the line can be used as a (the line doesn't care which one you start from).
The direction d can be replaced by any non-zero multiple of itself (2d, −d, ½d all point along the same line). So two equations that look different can describe the same line — match them by checking the directions are parallel and a shared point fits both.
IB-style question — write the vector equation
A line passes through the point P(1, −2, 4) and is parallel to the vector d = (3, 0, −1).
Write down a vector equation of the line.
Step by step
- Use the point's position vector as a and the given direction as d. Then r = a + λd.
- Substitute into r = a + λd.
Final answer
r = (1, −2, 4) + λ(3, 0, −1). (Any parallel direction, e.g. (−3, 0, 1), is equally correct.)
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Two points give you the direction for free: Given two points A and B on the line, the direction is just the displacement from one to the other:
d = AB = b − a (final position minus starting position).
Then take either point as the start. So the line through A and B is r = a + λ(b − a).
IB-style question — line through two points
A line passes through A(2, 1, 5) and B(4, 5, 3).
Find a vector equation of the line.
Step by step
- Direction = B − A (subtract coordinate by coordinate).
- Use A as the starting point a.
- You may simplify the direction by dividing by 2 (it stays parallel).
Final answer
r = (2, 1, 5) + λ(2, 4, −2), or equivalently with direction (1, 2, −1).
Parametric form: split into coordinates: Reading the vector equation one row at a time gives the parametric equations — one equation per coordinate, all sharing the same λ:
x = a₁ + λd₁, y = a₂ + λd₂, z = a₃ + λd₃.
These are the same line, just written as separate coordinate formulas. They're handy for substituting into other equations or eliminating λ.
IB-style question — write the parametric form
A line has vector equation r = (1, −2, 4) + λ(3, 0, −1).
Write down its parametric equations.
Step by step
- Take each row of the vector equation separately, keeping the same λ.
- Tidy the y-row (the direction's y-component is 0, so y is constant).
Final answer
x = 1 + 3λ, y = −2, z = 4 − λ.