medium difficulty10 min

Shortest Distance Between Skew and Parallel Lines

The shortest distance between two skew lines is the length of the common perpendicular between them. Skew lines are non-parallel, non-intersecting lines in three-dimensional space.

In one line: For skew lines, cross product finds the common perpendicular direction; dot product measures the gap along it.

Practice This Concept

Understand it

For lines r = a₁ + λb₁ and r = a₂ + μb₂, the vector a₂ - a₁ joins one point on the first line to one point on the second line. The vector b₁ × b₂ gives a direction perpendicular to both lines. Projecting a₂ - a₁ on this perpendicular direction gives the shortest distance.

Example

For r = i + 2j + 3k + λ(i - j + k) and r = 2i - j + k + μ(2i + j - k), use a₁ = (1,2,3), a₂ = (2,-1,1), b₁ = (1,-1,1), and b₂ = (2,1,-1) in the skew-lines distance formula.

Formula and rule

For skew lines r = a₁ + λb₁ and r = a₂ + μb₂, shortest distance = |(a₂ - a₁) · (b₁ × b₂)| / |b₁ × b₂|, with condition b₁ × b₂ ≠ 0.

For parallel lines, use distance = |(a₂ - a₁) × b| / |b|, where b is the common direction vector.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Find the shortest distance between r = (1,0,2) + λ(1,2,2) and r = (0,1,1) + μ(2,1,3).

1Here a₁ = (1,0,2), a₂ = (0,1,1), b₁ = (1,2,2), b₂ = (2,1,3). a₂ - a₁ = (-1,1,-1). b₁ × b₂ = |i j k; 1 2 2; 2 1 3| = i(6 - 2) - j(3 - 4) + k(1 - 4) = (4,1,-3).
2Dot product: (-1,1,-1) · (4,1,-3) = -4 + 1 + 3 = 0.
3Shortest distance = |0|/√(4²+1²+(-3)²) = 0.
4The lines intersect, so the shortest distance is 0.

Method you can reuse

1Convert both lines to vector form if needed.
2Identify a₁, a₂, b₁, and b₂.
3Compute b₁ × b₂.
4If it is zero, use the parallel-lines formula.
5If it is non-zero, compute a₂ - a₁, then the scalar triple product (a₂ - a₁) · (b₁ × b₂).
6Take the modulus and divide by |b₁ × b₂|.
7State the distance with correct units if given.

Visual tip

A 3D diagram should show two non-intersecting, non-parallel lines, their direction vectors, and the common perpendicular segment connecting them.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is b₁ × b₂ used in the shortest distance formula for skew lines?

Think first, then show answer

Because b₁ × b₂ is perpendicular to both line directions, so it gives the direction of the common perpendicular.

Common wrong answer

Using |a₂ - a₁| as the shortest distance without checking perpendicularity to both line directions.

Don't mix this up

Students often use the distance between the two given points on the lines, which is usually not the shortest distance unless that joining segment is perpendicular to both lines.

Exam tip

Before applying the skew-lines formula, check that b₁ × b₂ is not the zero vector. If it is zero, the lines are parallel and require the parallel-lines distance method.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Asked in long-answer form with two vector equations or Cartesian equations. The question may require identifying whether lines are skew, intersecting, or parallel before selecting the distance formula.

1
Enough for 1 mark
The shortest distance between two skew lines is the length of the common perpendicular between them. Skew lines are non-parallel, non-intersecting lines in three-dimensional space.
2
Add for 2 marks
Example: For r = i + 2j + 3k + λ(i - j + k) and r = 2i - j + k + μ(2i + j - k), use a₁ = (1,2,3), a₂ = (2,-1,1), b₁ = (1,-1,1), and b₂ = (2,1,-1) in the skew-lines distance formula.
3
Add for 3 marks
For lines r = a₁ + λb₁ and r = a₂ + μb₂, the vector a₂ - a₁ joins one point on the first line to one point on the second line.

MCQ Quiz

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