Relationships Between Lines (HL only)
Relationships Between Lines (HL only)
For \(l_1:\mathbf r=\mathbf a+\lambda\mathbf u\) and \(l_2:\mathbf r=\mathbf b+\mu\mathbf v\), the direction vectors must be nonzero.
Parallel lines
Skew lines
Intersecting lines
Acute angle between lines
| Direction test | Position test | Conclusion |
|---|---|---|
| Parallel directions | One point lies on both lines | Coincident |
| Parallel directions | No common point | Distinct parallel |
| Nonparallel directions | One pair of parameters satisfies every component | Intersecting |
| Nonparallel directions | No pair satisfies every component | Skew in three dimensions |
Solve \(\mathbf a+\lambda\mathbf u=\mathbf b+\mu\mathbf v\) using two independent component equations where possible, then check every remaining component equation. A solution to only two equations does not establish an intersection.
The smaller angle \(\alpha\) between lines satisfies \(\cos\alpha=\dfrac{|\mathbf u\cdot\mathbf v|}{|\mathbf u||\mathbf v|}\), with \(0\le\alpha\le\dfrac\pi2\).
Parallel and skew are different: skew lines have different directions and no common point. Keep independent parameters for the two lines.
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