Skew lines: intersecting, parallel or skew?
Skew lines are lines in space that are neither parallel nor intersecting. They do not lie in one plane, so the shortest distance between them is never zero.
Three ways two lines can sit in space
In a plane, two lines either meet or are parallel. In space there is a third way. Two lines can point in different directions and still never meet, because one passes above the other. Such lines are skew lines. They do not lie in any one plane. Think of a flyover: the upper road and a road below running in another direction never meet, and they are not parallel.
Parallel, intersecting or skew
| Check | Parallel | Intersecting | Skew |
|---|---|---|---|
| DRs proportional? | Yes | No | No |
| Common point? | None (every point if the lines coincide) | Exactly one | None |
| Lie in one plane? | Yes | Yes | No |
| Shortest distance | A constant gap (0 if they coincide) | 0 | Positive |
How to test two lines
- 1Compare the DRs. If they are proportional, the lines are parallel (or the same line).
- 2Otherwise, write a general point of each line, one with λ and one with μ.
- 3Equate the x-coordinates and the y-coordinates. Solve these two equations for λ and μ.
- 4Put λ and μ into the z-coordinates.
- 5★If the z-coordinates are equal, the lines intersect: find the point. If not, the lines are skew.
★ key step
Example 1: lines that meet
Show that the lines (x − 1)/1 = (y − 2)/(−1) = (z − 3)/2 and (x − 2)/1 = (y + 2)/2 = (z − 8)/(−1) intersect, and find the point.
- General points: (1 + λ, 2 − λ, 3 + 2λ) and (2 + μ, −2 + 2μ, 8 − μ)
- x: 1 + λ = 2 + μ, so λ − μ = 1
- y: 2 − λ = −2 + 2μ, so λ + 2μ = 4
- Subtract: 3μ = 3, so μ = 1 and λ = 2
- z: 3 + 2(2) = 7 and 8 − 1 = 7. Equal, so the lines meet.
- Point: (1 + 2, 2 − 2, 7) = (3, 0, 7)
The lines intersect at (3, 0, 7).
Example 2: skew lines
Show that the lines (x − 1)/2 = y/1 = (z + 1)/(−1) and x/1 = (y − 2)/1 = (z − 1)/2 are skew.
- DRs 2, 1, −1 and 1, 1, 2 are not proportional, so the lines are not parallel.
- General points: (1 + 2λ, λ, −1 − λ) and (μ, 2 + μ, 1 + 2μ)
- x: 1 + 2λ = μ; y: λ = 2 + μ
- Substitute: λ = 2 + 1 + 2λ, so λ = −3 and μ = −5
- z: −1 − (−3) = 2, but 1 + 2(−5) = −9
- 2 ≠ −9, so there is no common point.
The lines are neither parallel nor intersecting, so they are skew.
How to draw skew lines
- Draw a cuboid (a room) ABCD-EFGH, with ABCD the floor and E, F, G, H directly above A, B, C, D.
- Darken the floor edge AB and label it.
- Darken the vertical edge CG and label it.
- Write: AB is horizontal and CG is vertical, so they are not parallel; they never meet, so they are skew.
- Mark the edge BC: it is perpendicular to both, and its length is the shortest distance between AB and CG.
On a flyover, the upper road and a road below running in another direction behave like skew lines. Planners need the shortest gap between them to fix the safe height for buses and trucks.
1 marks: “Define skew lines.” Lines in space that are neither parallel nor intersecting. They do not lie in the same plane.
2 marks: “Are the lines r⃗ = (î − ĵ) + λ(2î + k̂) and r⃗ = (2î − ĵ) + μ(î + ĵ − k̂) skew?” DRs 2, 0, 1 and 1, 1, −1 are not proportional. Points: (1 + 2λ, −1, λ) and (2 + μ, −1 + μ, −μ). y gives μ = 0, x gives λ = 1/2. z: 1/2 ≠ 0. No common point, so the lines are skew.
3 marks: “Show that the lines (x + 1)/2 = (y − 1)/1 = (z − 4)/(−1) and x/1 = (y − 4)/(−2) = z/3 intersect. Find the point of intersection.” Points (−1 + 2λ, 1 + λ, 4 − λ) and (μ, 4 − 2μ, 3μ). x: μ = −1 + 2λ; y: 1 + λ = 4 − 2μ = 6 − 4λ, so λ = 1 and μ = 1. z: 3 and 3 match. Point (1, 2, 3).
Skew = not parallel, never meet, not in one plane.