AHL 3.15 (HL)—Line relationships in 3D

Classify 3D lines by solving their parameter conditions.

Two lines may intersect, be parallel, or be skew; comparing direction vectors and solving all three coordinate equations distinguishes the cases.

Example

If a+λd=b+μe has one consistent pair (λ,μ), the lines intersect; if d is a scalar multiple of e they are parallel or coincident.

Solve coordinates together rather than checking one projection, then verify the resulting point in all three components.

Crossing in an x–y sketch does not prove a 3D intersection; skew lines can have matching projections.