Relationships Between Lines (HL only)

For l_1: r= a+lambda u and l_2: r= b+mu 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 a+lambda u= b+mu 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 cosalpha= | u. v|| u|| v|, with 0 <= alpha <= 2. Parallel and skew are different: skew lines have different directions and no common point. Keep independent parameters for the two lines.