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Learn7367Paper 1 · Compulsory contentFurther vectorsIntersecting, parallel and skew lines
Further vectors · Subtopic 3 of 4

Intersecting, Parallel and Skew Lines

In a plane, two lines either cross or are parallel. In three dimensions they can also miss each other completely. Learn the three-step test that decides, and how to find the crossing point when there is one.

Cambridge 9709 Paper 3Edexcel YMA01 WMA14AQA 7367 Paper 1≈ 40 minRevision 5 · updated 2026-10-27
Section 1 of 14

Learning objectives

  • Decide whether two lines are parallel by comparing direction vectors.
  • Solve two component equations for λ\lambda and μ\mu, and use the third to decide whether the lines meet.
  • Find the point of intersection, or show that the lines are skew.
  • Find an unknown constant that makes two lines intersect.
Pause and think

A road runs north–south at ground level. A bridge carries another road east–west, 66 m up. Looking from above, the roads cross. Do they meet?

Section 2 of 14

Core idea

Start hereDo the flight paths cross?

Two planes fly straight lines at different heights. In 2D, non-parallel lines always meet. In 3D they might pass over and under each other without ever touching: skew lines. Deciding which needs three equations and only two unknowns, so one equation is a check.

The big idea

Lines are parallel (directions are multiples), intersecting, or skew. To test: set the position vectors equal; solve two component equations for λ,μ\lambda,\mu; substitute into the third. Consistent ⇒ intersect; inconsistent ⇒ skew.

1. The method

Classifying two lines

Three equations, two unknowns: the third is the test.

Worked exampler=(1,2,3)+λ(1,−1,2)\mathbf r=(1,2,3)+\lambda(1,-1,2), r=(3,2,4)+μ(1,1,−1)\mathbf r=(3,2,4)+\mu(1,1,-1)

Do they meet?

Goal
Point or "skew".
Strategy
Equate components.
  1. 1+λ=3+μ,2−λ=2+μ,3+2λ=4−μ1+\lambda=3+\mu,\quad2-\lambda=2+\mu,\quad3+2\lambda=4-\mu
    xx, yy, zz.
  2. μ=−λ\mu=-\lambda
    From the second.
  3. 1+λ=3−λ⇒λ=1, μ=−11+\lambda=3-\lambda\Rightarrow\lambda=1,\ \mu=-1
    Into the first.
  4. 3+2=5, 4+1=5 ✓3+2=5,\ 4+1=5\ \checkmark
    Third equation consistent.
AnswerThey meet at (2,1,5)(2,1,5).
✓ CheckSecond line at μ=−1\mu=-1: (3−1,2−1,4+1)=(2,1,5)(3-1,2-1,4+1)=(2,1,5) ✓.
Skew lines along the edges of a box
L₁ (along the floor)L₂ (across the roof)
L1L_1 runs along the bottom front edge; L2L_2 runs front-to-back across the top. Different directions, different heights: they never meet.
Worked exampler=(1,2,3)+λ(1,−1,2)\mathbf r=(1,2,3)+\lambda(1,-1,2), r=(0,0,1)+μ(2,1,0)\mathbf r=(0,0,1)+\mu(2,1,0)

Classify.

Goal
Relationship.
Strategy
Same test.

  1. Directions not multiples: not parallel.

  2. 1+λ=2μ,2−λ=μ,3+2λ=11+\lambda=2\mu,\quad2-\lambda=\mu,\quad3+2\lambda=1
    Components.
  3. λ=−1, μ=3\lambda=-1,\ \mu=3
    From the third and second.
  4. 1+(−1)=0≠61+(-1)=0\neq6
    First equation fails.
AnswerSkew.
✓ CheckNot parallel and no common point ✓.
Pause and think

Are directions (1,2,−1)(1,2,-1) and (−2,−4,2)(-2,-4,2) parallel?

Summary

  • Parallel check first.
  • Solve two, test the third.
Key words
Skew lines
Non-parallel lines that never meet.
NoteThe decision in three steps. 1. Are the directions parallel? If yes: same line or parallel. 2. If not, solve two component equations for λ\lambda and μ\mu. 3. Check the third: true means they intersect, false means skew.
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