Parallel, Intersecting and Skew Lines — Pure Mathematics 3 (9709) — Mathematics

Introduction

In two dimensions, two distinct straight lines must either be parallel or they intersect — there is no third option. In three dimensions, however, a third possibility arises: two lines can be skew, meaning they are neither parallel nor do they meet. This subtopic equips you to classify any pair of lines in 3D and, when an intersection exists, to find the exact point. It appears regularly in Paper 3 and is usually worth 6–9 marks; a systematic algebraic approach is essential for full credit.

Core Concept

Recall that the vector equation of a line through point $\mathbf{a}$ with direction $\mathbf{d}$ is:

\mathbf{r} = \mathbf{a} + \lambda \mathbf{d}

where $\lambda \in \mathbb{R}$ is a scalar parameter.

Given two lines $\ell_1$ and $\ell_2$ :

\ell_1: \mathbf{r} = \mathbf{a} + \lambda \mathbf{d}_1 \qquad \ell_2: \mathbf{r} = \mathbf{b} + \mu \mathbf{d}_2

there are exactly three possible relationships.

Step 1 — Check for Parallelism

The lines are parallel if and only if their direction vectors are scalar multiples of each other:

\mathbf{d}_1 = k\,\mathbf{d}_2 \quad \text{for some } k \in \mathbb{R}

If $\mathbf{d}_1$ and $\mathbf{d}_2$ are not parallel, proceed to Step 2.

Step 2 — Test for Intersection

Set the position vectors equal and attempt to solve simultaneously:

\mathbf{a} + \lambda \mathbf{d}_1 = \mathbf{b} + \mu \mathbf{d}_2

This gives three scalar equations (one per component: $x$ , $y$ , $z$ ) in two unknowns ( $\lambda$ and $\mu$ ). The system is overdetermined.

Choose any two equations and solve for $\lambda$ and $\mu$ .
Substitute the values into the third equation.
- If the third equation is satisfied ✓ — the lines intersect; substitute back to find the point.
- If the third equation is not satisfied ✗ — the lines are skew.

Note: The syllabus does not require the shortest distance between skew lines or the common perpendicular.

Classification Summary

Directions parallel?	Third equation satisfied?	Classification
Yes (and lines coincide)	—	Same line (coincident)
Yes (and lines distinct)	—	Parallel
No	Yes	Intersecting
No	No	Skew

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