The four cases

In brief — How to distinguish the four cases

Given $r_1:\vec S=\vec A+\vec v_1 t$ and $r_2:\vec S=\vec B+\vec v_2 s$:

1. Is $\vec v_1\parallel\vec v_2$? If yes the lines are parallel (or coincident: check whether a point of $r_1$ belongs to $r_2$).
2. If $\vec v_1\not\parallel\vec v_2$ set up the system $\vec A+\vec v_1 t = \vec B+\vec v_2 s$ (3 equations, 2 unknowns). From two equations one obtains $t,s$; the third acts as a check: if it is satisfied the lines are incident, otherwise they are skew.

Skew lines — neither parallel nor incident — are the real novelty of space: they do not exist in the plane.

The three non-trivial cases: parallel, incident and skew lines.

Links

Topics: Analytic geometry in space
Concepts: Line · Line in space · Skew lines
Skills: Analytic geometry · Reasoning by cases · Solving systems