The Shortest Distance Between Two Skew Lines in ℝ³

The Shortest Distance Between Two Skew Lines in ℝ³

This post derives, from first principles, a vector formula for the shortest distance between two skew lines in ℝ³. The argument uses only the definitions and basic properties of the dot product and cross product; no higher results are assumed.

1. Vector Equations of the Lines

Let the two lines be given in vector form by 

r = a + λb
r = c + μd

where:

  • a, c are position vectors of fixed points on each line,
  • b, d are non-zero direction vectors,
  • λ, μ ∈ ℝ are parameters.

The lines are skew if they are neither parallel nor intersecting. Our goal is to find a closed-form expression for the minimum distance between them.

2. Direction of the Common Perpendicular

The segment that realises the shortest distance lies along a line perpendicular to both direction vectors b and d. A vector perpendicular to both is given by their cross product: 

b × d.

Assuming b and d are not parallel, b × d ≠ 0. A unit vector in the direction of the common perpendicular is therefore 

u = (b × d) / ‖b × d‖.

3. A General Joining Vector Between the Lines

Let P be a point on the first line and Q a point on the second: 

P = a + λb,
Q = c + μd.

The vector from Q to P is 

QP = P − Q
   = (a + λb) − (c + μd)
   = (a − c) + λb − μd.

For different choices of λ and μ we obtain different joining vectors, but the shortest distance will be the component of any such vector in the direction of the common perpendicular.

4. Projection of QP onto the Common Perpendicular

The signed length of the projection of QP onto the unit vector u is 

QP · u = QP · (b × d) / ‖b × d‖.

Substituting QP gives 

QP · (b × d)
= (a − c + λb − μd) · (b × d)
= (a − c) · (b × d)
  + λ (b · (b × d))
  − μ (d · (b × d)).

5. Orthogonality and Independence of λ, μ

For any vectors u, v we have u · (u × v) = 0, since u × v is perpendicular to u. Applying this to b and d gives 

b · (b × d) = 0,   d · (b × d) = 0.

Hence the terms involving λ and μ vanish, and we are left with 

QP · (b × d) = (a − c) · (b × d).

This shows that the component of QP in the direction of the common perpendicular does not depend on which points P and Q we choose on the two lines: all choices lead to the same value.

6. The Distance Formula

The signed projection of QP onto the common perpendicular is 

(a − c) · (b × d) / ‖b × d‖.

The shortest distance between the two skew lines is the absolute value of this projection: 

Distance = |(a − c) · (b × d)| / ‖b × d‖.

The absolute value is required because distance is always non-negative.

7. Final Statement

For two skew lines in ℝ³ with vector equations 

r = a + λb,
r = c + μd,

where b and d are non-parallel direction vectors, the shortest distance between the lines is 

Distance = |(a − c) · (b × d)| / ‖b × d‖.

This formula measures the component of the vector joining the two lines in the unique direction perpendicular to both. It is obtained purely from the definitions of the dot product and cross product and the elementary geometry of projections in ℝ³.

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