Shortest Distance Between Skew Lines: Formula & Examples

Skew lines are lines in 3D that are neither parallel nor intersecting — they pass by each other like two flyovers at different heights. The shortest distance between them is the length of the unique segment that is perpendicular to both lines.

There is exactly one such common perpendicular, so this distance is well-defined. The formula uses the scalar triple product: if the lines are r = a1 + λd1 and r = a2 + μd2, then distance = |[d1 d2 (a2 − a1)]| / |d1 × d2|.

Key Concepts

• Skew lines: non-parallel lines that do not intersect. In 2D this is impossible; in 3D it is common.
• Common perpendicular: the single line segment perpendicular to both lines; its length is the shortest distance between them.
• The numerator |[d1 d2 (a2 − a1)]| is the volume of the box spanned by the two direction vectors and the vector joining a point of one line to a point of the other.
• The denominator |d1 × d2| is the area of the base parallelogram; volume divided by base area equals height, which is exactly the distance.
• The formula needs d1 × d2 ≠ 0, that is, the lines must not be parallel. For parallel lines use distance = |(a2 − a1) × d| / |d| instead.
• Units: the distance is measured in the same length units as the coordinates.

Formulas

• d = |(d1 × d2) · (a2 − a1)| / |d1 × d2|.
• For parallel lines with direction d: d = |(a2 − a1) × d| / |d|.
• If the numerator is zero, the lines intersect and the distance is 0.

Worked Example

Find the shortest distance between L1: r = (1, 0, 0) + λ(1, 1, 0) and L2: r = (0, 0, 2) + μ(0, 1, 1).

1. d1 = (1, 1, 0), d2 = (0, 1, 1), and a2 − a1 = (0 − 1, 0 − 0, 2 − 0) = (−1, 0, 2).
2. d1 × d2: x = 1×1 − 0×1 = 1; y = 0×0 − 1×1 = −1; z = 1×1 − 1×0 = 1. So n = (1, −1, 1).
3. |n| = √(1 + 1 + 1) = √3.
4. Numerator: |n · (a2 − a1)| = |1×(−1) + (−1)×0 + 1×2| = |−1 + 0 + 2| = 1.
5. Distance = 1/√3 ≈ 0.577 units.
6. Confirm skew: d1 × d2 ≠ 0 means not parallel, and the non-zero numerator means they do not intersect. Correct.

Common Misconceptions

• The skew-lines formula works for parallel lines too. It does not — the denominator |d1 × d2| becomes zero. Parallel lines need the point-to-line formula.
• Any perpendicular segment between the lines is the shortest. Only the unique common perpendicular gives the minimum; every other connecting segment is longer.
• Skew lines are just lines that do not intersect. Parallel lines also do not intersect; skew specifically means non-parallel AND non-intersecting.

Real-World Applications

• Air-traffic control: minimum separation between flight paths.
• Mechanical design: clearances between shafts, pipes and cables.
• Robotics: keeping moving arms from colliding with each other.
• Architecture: spacing of non-parallel structural members.

For skew lines r = a1 + λd1 and r = a2 + μd2, the shortest distance is the triple-product magnitude divided by |d1 × d2|. Check non-parallel first, and switch formulas if the lines turn out parallel.