Vector equations of lines
Learning intention
Form and interpret vector equations of lines, then use parameters to test intersections and geometric relationships.
The core idea
A line in two or three dimensions is determined by one point and one non-zero direction vector. Its vector equation generates every point on the line by changing a real parameter.
Two lines intersect only if the same point can be produced by both equations. This creates simultaneous equations in separate parameters; every coordinate equation must be satisfied.
Key definitions and formulas
- Vector line: r = a + λd, where a is a position vector and d is a direction vector.
- Parallel lines: their direction vectors are scalar multiples.
- Intersection: solve a + λd = b + μe consistently in every coordinate.
- Coincident lines: parallel directions and a point from one line lying on the other.
Worked example 1 — foundation
Find a vector equation through A(1, 2, −1) and B(4, 0, 5).
- Use A as the fixed point: a = (1, 2, −1).
- Find direction B − A = (3, −2, 6).
- Introduce a real parameter λ.
r = (1, 2, −1) + λ(3, −2, 6).
Worked example 2 — exam-style
Test whether P(7, −2, 11) lies on r = (1, 2, −1) + λ(3, −2, 6).
- From x: 1 + 3λ = 7, so λ = 2.
- Check y: 2 − 2(2) = −2.
- Check z: −1 + 6(2) = 11.
- All coordinates use the same parameter.
P lies on the line at λ = 2.
Worked example 3 — HL reasoning
Lines have directions d = (1, 2, −1) and e = (2, 4, −2). Classify their relationship before testing points.
- Observe e = 2d.
- The lines are parallel or coincident; they cannot meet at exactly one point.
- Substitute one fixed point into the other line to distinguish the two cases.
Direction alone proves parallel orientation, but fixed points decide parallel distinct versus coincident.
Common IB mistake
Solving only the x- and y-coordinate equations is not enough in three dimensions. The z-coordinate may make the system inconsistent, which is how skew lines can be detected.
Practice checks
Use the feedback to refine your method, not only your final answer.
Summary
- Use point plus parameter times direction.
- Subtract points to build a direction vector.
- Use one consistent parameter for point membership.
- Check every coordinate when solving intersections.
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