Distance between skew lines
Two aeroplane routes cross over each other at different heights. They never meet, yet there is a smallest gap between them. This gap lies along the common perpendicular.
Lines in space are skew if they are not parallel and do not meet. For r = a₁ + λb₁ and r = a₂ + μb₂, the shortest distance is the projection of a₂ − a₁ on the vector b₁ × b₂, which is perpendicular to both lines. If the distance is zero, the lines intersect.
Formula: d = |(b₁ × b₂) · (a₂ − a₁)| / |b₁ × b₂|
Rule 1: Cross product, dot product, divide — Work in three steps.
- Easy: The x-axis and the line through (0, 0, 1) in the direction of j: b₁ × b₂ = k, a₂ − a₁ = k, so d = 1.
- Medium: Lines (x − 1)/1 = (y − 1)/0 = z/1 and x/0 = y/1 = (z − 1)/0 give b₁ × b₂ = −i + k, and d = 2/√2 = √2.
- Exam-type: The line through the origin along i + j + k and the line through (1, 0, 1) along j meet at (1, 1, 1), because the numerator becomes 0.
Rule 2: More skew pairs — Keep the same steps.
- Easy: The x-axis and the line through (0, 3, 4) along j are at distance 4.
- Medium: The line through the origin along i and the line through (1, 1, 5) along j + k are at distance 2√2.
- Exam-type: The line through the origin along i + j + k and the line through (1, 0, 0) along i + j are at distance 1/√2.
The numerator is a triple product. If it is zero, the lines are in a common plane and they meet.
Forgetting to divide by |b₁ × b₂|.
Common perpendicular direction is the cross product of the two directions.
Practice questions
- 1Find the shortest distance between the x-axis and the line through (0, 0, 1) along j.
- 2Find d for (x − 1)/1 = (y − 1)/0 = z/1 and x/0 = y/1 = (z − 1)/0.
- 3Do the line through the origin along i + j + k and the line through (1, 0, 1) along j intersect?
- 4Find the shortest distance between the z-axis and the line through (1, 0, 0) along j.
- 5Find the shortest distance between the x-axis and the line through (0, 3, 4) along j.
- 6Find d between the line through the origin along i and the line through (1, 1, 5) along j + k.
- 7Write the formula for the shortest distance between skew lines.
- 8Find d between the line through the origin along i + j + k and the line through (1, 0, 0) along i + j.
Show answers (try them yourself first)
- 11
- 2√2
- 3Yes, the numerator is 0, so they meet at (1, 1, 1)
- 4k × j = −i, a₂ − a₁ = i, so d = 1
- 54
- 64/√2 = 2√2
- 7d = |(b₁ × b₂) · (a₂ − a₁)| / |b₁ × b₂|
- 81/√2
Distance between parallel lines
Two parallel pipes run side by side at a fixed gap. The gap is the same everywhere along their length, and it is found from the cross product of the common direction with the vector joining two points.
If two lines are parallel with the same direction b, then the distance between them is the length of the perpendicular from a point of one line to the other. The vector between the points is a₂ − a₁.
Formula: d = |b × (a₂ − a₁)| / |b|
Rule 1: Cross with the common direction — Divide by its length.
- Easy: r = λi and r = j + μi: d = |i × j|/1 = 1.
- Medium: Through the origin along i + j and through (1, 0, 0) along 2(i + j): d = 1/√2.
- Exam-type: Through the origin along 3i + 4j and through (0, 0, 5) along the same: b × (a₂ − a₁) = 20i − 15j, so d = 25/5 = 5.
Rule 2: Axis-parallel lines — Use the plane distance.
- Easy: The x-axis and the line y = 3, z = 4 are 5 units apart.
- Medium: The lines x = 1, y = 2 and x = 4, y = 6 (both parallel to the z-axis) are 5 apart.
- Exam-type: r = λ(i + j + k) and r = i + μ(i + j + k) are √6/3 apart.
Parallel lines have the same direction, so b₁ is a multiple of b₂. Then use b × (a₂ − a₁).
Using the skew line formula, which gives zero because b₁ × b₂ = 0.
Cross the direction with the gap vector, then divide.
Practice questions
- 1Find the distance between r = λi and r = j + μi.
- 2Find the distance between the line through the origin along i + j and the line through (1, 0, 0) along 2i + 2j.
- 3Find the distance between the line through the origin along 3i + 4j and the line through (0, 0, 5) with the same direction.
- 4Find the distance between the x-axis and the line y = 3, z = 4.
- 5Find the distance between r = λ(i + j + k) and r = i + μ(i + j + k).
- 6Find the distance between the lines x = 1, y = 2 and x = 4, y = 6, both parallel to the z-axis.
- 7What is the condition for two lines to be parallel?
- 8Find the distance between r = λ(2i) and r = 3k + μi.
Show answers (try them yourself first)
- 11
- 21/√2
- 35
- 4√(9 + 16) = 5
- 5√6/3, since b × i = j − k has length √2 and |b| = √3
- 6√(9 + 16) = 5
- 7Their direction vectors are proportional, b₁ = k b₂
- 83