The angle between two lines, and parallel, perpendicular, intersecting and skew lines
The angle between two lines is the angle between their direction vectors, but a line has two directions, d and −d, so we take the acute (or right) one: cos θ = |d₁ · d₂|/(‖d₁‖‖d₂‖), with 0° ≤ θ ≤ 90°. The absolute value in the top does this. If the lines are given by direction cosines, then cos θ = |ℓ₁ℓ₂ + m₁m₂ + n₁n₂|. Special cases: the lines are perpendicular when d₁ · d₂ = 0, and parallel when d₁ = kd₂ (or d₁ × d₂ is the zero vector). Parallel lines that share a point are the same line. In a plane, two lines that are not parallel must meet. In space there is a third possibility. Two lines that are not parallel either intersect in exactly one point, or they are skew (in Arabic متخالفان): they never meet and are not parallel, for example an edge on the floor of a room and a different-direction edge on its ceiling. To decide, give the lines different parameters, s for the first and t for the second. Equate the coordinates, which gives three equations in two unknowns. Solve two of them, then test the third: if it is satisfied the lines meet at the point found, and if not they are skew. Perpendicular lines may meet or may be skew; perpendicularity only compares the directions.
| directions | common point? | the two lines are |
|---|---|---|
| proportional: d₁ = kd₂ | none, or all points | parallel (or the same line) |
| not proportional | exactly one | intersecting |
| not proportional | none | skew |
Edges of a room. Stand in a room and think of three edges: one on the floor along the east wall, one on the floor along the north wall, and one on the ceiling along the east wall. Which pairs are perpendicular and meet, parallel, or skew? (Floor-east and floor-north meet and are perpendicular; floor-east and ceiling-east are parallel; ceiling-east and floor-north are perpendicular and skew.)