A Level · Cambridge (CIE) · 9709

Classify parallel intersecting and skew lines with independent parameters

Compare directions, solve the coordinate equations and check the third dimension before claiming an intersection.

Section 3.7 · Vectors

Pure Mathematics 3 · Vectors · 3.7

Exam scope and assessment notes

Pure Mathematics 3 (Paper 3, A Level) · Mathematics 9709, 2026–2027, section 3.7. Paper 1 coordinate geometry and algebra are assumed. Work in two or three dimensions as stated. The general section theorem is not included; midpoint geometry is required. The cross product, shortest distance between skew lines and their common perpendicular are not required. Scalar products include angles between lines and the foot of a perpendicular from a point to a line, including 3D solids. Written practice is self-checked; human subject-teacher review remains pending.

Before you begin

Interpret a vector line equation and test point membership.
Solve two simultaneous linear equations and substitute into a third.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

Which vector is proportional to (1, −1, 2)ᵀ?

What you will learn

Distinguish distinct parallel, coincident, intersecting and skew lines in three dimensions.
Use independent parameters to find an intersection and separate path crossing from simultaneous meeting in a timed model.

Compare directions before solving for a crossing

For r = a + td and r = b + se, use independent real parameters t and s. The direction vectors are parallel if e = kd for some non-zero constant k; check every component, including zeros. Parallel directions can describe distinct parallel lines or one coincident line. If b − a is also a scalar multiple of d, the anchors belong to the same line and the lines coincide, giving infinitely many common points rather than one intersection.

For a = (1, 2, 3)ᵀ and d = (1, −1, 2)ᵀ, the line with anchor (0, 3, 1) and direction 2d is coincident: its anchor equals a − d. The line with anchor (1, 2, 4) and direction 2d is instead distinct and parallel, because its offset (0, 0, 1) is not proportional to d. Different anchors alone do not prove two parallel lines are distinct.

A projected crossing is not a 3D intersection

For non-parallel directions, solve the coordinate equations for t and s and then verify every remaining coordinate. The lines (1, 0, 2)ᵀ + t(2, 1, −1)ᵀ and (2, 5, −3)ᵀ + s(1, −1, 1)ᵀ give t = 2 and s = 3 from x and y. Both z values are then 0, so the intersection is (5, 2, 0). There is no reason for the two parameters to be equal. They describe separate positions along separate lines.

Changing only the second anchor's z coordinate from −3 to −2 leaves the x and y equations unchanged, but at t = 2 and s = 3 the z coordinates become 0 and 1. These non-parallel lines have no common point, so they are skew. Skew lines occur in 3D; their 2D projections may cross. The syllabus does not require their shortest distance or common perpendicular. If parameters represent elapsed time with a shared start, a spatial crossing at different times does not establish a simultaneous meeting.

Pause and explain

At a candidate intersection, the first line has t = 2 and the second has s = 3. Must the claim fail because t ≠ s?

Put the idea to work

Worked example

Find the intersection of r = (1, 0, 2)ᵀ + t(2, 1, −1)ᵀ and r = (2, 5, −3)ᵀ + s(1, −1, 1)ᵀ.

Show the worked solution

Equate x: 1 + 2t = 2 + s; equate y: t = 5 − s.
Solve to get t = 2 and s = 3, then check z: 2 − 2 = −3 + 3 = 0.
Substitute into either whole vector equation to obtain the same point (5, 2, 0).

Answer Intersection (5, 2, 0), with t = 2 and s = 3.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

Classify r = (1, 2, 3)ᵀ + t(1, −1, 2)ᵀ and r = (0, 3, 1)ᵀ + s(2, −2, 4)ᵀ as distinct parallel or coincident lines.

Give me a hint

Compare directions, then test whether the second anchor lies on the first line.

Compare my reasoning

The second direction is twice the first, so their directions are parallel.
The anchor difference is (−1, 1, −2)ᵀ, equal to −1 times the first direction.
The lines coincide; t = 2s − 1 relates their equations, giving infinitely many common points.

Coincident lines, with parameter relation t = 2s − 1.

Look for these in your work

Check every direction component with one multiplier.
Test the anchor offset before claiming the lines are distinct.

2 · Independent

Classify r = (0, 0, 0)ᵀ + t(1, 1, 0)ᵀ and r = (1, 0, 1)ᵀ + s(−1, 1, 0)ᵀ, showing all coordinate checks.

Give me a hint

Solve x and y, then compare the constant z values.

Compare my reasoning

The directions are not proportional. Equating x and y gives t = 1 − s and t = s, hence t = s = 1/2.
The first line's z is 0 and the second line's z is 1 at every parameter value.
There is no 3D intersection, so these non-parallel lines are skew despite the projected crossing (1/2, 1/2).

Skew lines; the projected crossing has unequal z coordinates 0 and 1.

Look for these in your work

Check non-parallel directions and solve independent parameters.
Use the third coordinate to decide actual 3D intersection.

3 · Transfer

Two simulated particles start simultaneously. Their positions in metres are p(t) = (0, 0, 0)ᵀ + t(1, 0, 1)ᵀ and q(s) = (2, −1, 1)ᵀ + s(0, 1, 1)ᵀ, where t and s are their elapsed seconds. Do their forward paths intersect, and does the model predict a simultaneous meeting?

Give me a hint

First solve the spatial equations independently; then check the shared-time requirement.

Compare my reasoning

The x and y equations give t = 2 and s = 1; the z values are both 2.
The forward paths therefore intersect at (2, 0, 2), reached at different elapsed times.
A simultaneous meeting requires t = s. The unique crossing has 2 ≠ 1, so these simultaneously started particles never meet in the supplied model.

Paths intersect at (2, 0, 2); no simultaneous meeting because the crossing times are 2 s and 1 s.

Look for these in your work

Separate spatial intersection from a shared-time condition.
Check forward parameter ranges and all coordinates before interpreting the model.

Common mistakes

Independent line parameters need not have equal values at a common point.
Proportional directions require a separate anchor check to distinguish coincident from distinct parallel lines.
An x–y crossing can still fail the z equation and give skew lines.

Recall without your notes

What can you explain now?

Non-parallel 3D lines have a projected crossing, but their z values disagree there. How are they classified?

Compare with the explanation

Skew.

They are non-parallel and have no common 3D point; the projected crossing is insufficient.

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Make it stick

Classify two vector lines by their directions and coordinate equations. Explain coincident and skew cases, independent parameters, and why crossing paths do not always mean a simultaneous meeting.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.