Angle and shortest distance between lines

The essentials

The angle between two lines is the angle between their direction vectors. If the lines are parallel to b₁ and b₂, then cos θ = |b₁ · b₂|/(|b₁||b₂|). With direction ratios a₁, b₁, c₁ and a₂, b₂, c₂, cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / [√(a₁² + b₁² + c₁²) √(a₂² + b₂² + c₂²)]. The lines are perpendicular when a₁a₂ + b₁b₂ + c₁c₂ = 0 and parallel when the direction ratios are proportional.

In space, two lines can intersect, be parallel, or be skew: neither parallel nor intersecting (they lie in different planes). For skew lines r = a₁ + λb₁ and r = a₂ + μb₂, the shortest distance is measured along the common perpendicular: d = |(b₁ × b₂) · (a₂ − a₁)| / |b₁ × b₂|.

Key points

• If the shortest distance comes out as 0, the lines intersect.
• For parallel lines r = a₁ + λb and r = a₂ + μb, the distance is |b × (a₂ − a₁)| / |b|.
• The common perpendicular direction is b₁ × b₂.
• Example: the x-axis direction (1, 0, 0) and the direction (0, 1, 1) give cos θ = 0, so the lines are perpendicular.

Exam tips

• Write a₁, b₁, a₂, b₂ clearly before using the formula; most errors come from mixing them up.
• Use the modulus so that the angle is acute and the distance is positive.

Common mistakes

• Applying the skew-line formula to parallel lines, where b₁ × b₂ = 0 and the formula breaks down.
• Using position vectors of points instead of direction vectors when finding the angle.