IB Maths AA HL Topic 3 — Geometry & Trigonometry Paper 1 & 2 ~7 min read HL only

Coincident, Parallel, Intersecting & Skew Lines

In 3D, two lines can do four things: coincide (sit on top of each other), be parallel, intersect at a point, or be skew (neither parallel nor crossing). The classification rests on two checks — direction vectors, then a point.

📘 What you need to know

Parallel: direction vectors are scalar multiples (b₁ = kb₂).
Coincident: parallel AND any point on one line lies on the other.
Intersecting: not parallel; one value of λ and one of μ satisfies all three component equations.
Skew: not parallel AND don’t intersect — only possible in 3D.
Test for skew vs intersecting: solve two component equations for λ, μ; check the third — if consistent, intersect; if not, skew.
Intersection point: substitute λ (or μ) back into its line equation.
Use different parameters (λ and μ) — they’re independent variables, one per line.

The four cases

Case	Direction vectors	Common point?
Coincident	scalar multiples	infinitely many (same line)
Parallel	scalar multiples	none
Intersecting	not scalar multiples	exactly one
Skew	not scalar multiples	none

Decision flow: first ask “are the directions parallel?” If yes → coincident or parallel (test a point). If no → intersecting or skew (test the system of equations).

Step 1 — directions parallel?

Parallel test b₁ = kb₂ for some scalar k

Compare the components in pairs: x-ratio, y-ratio, z-ratio. If all three are equal to one constant k → parallel. If any disagrees → not parallel.

If parallel, take any point from one line (its anchor a₁) and check whether it lies on the other line. Lies on it → coincident; doesn’t → strictly parallel.

Step 2 — intersect or skew?

For non-parallel lines, set the two vector equations equal (with different parameters λ and μ):

Solve componentwise a₁ + λb₁ = a₂ + μb₂

This gives three linear equations in λ and μ. Solve any two — then verify the values satisfy the third. Consistent → lines intersect; inconsistent → skew.

🧭 Recipe — classify two lines in 3D

Compare direction vectors: are they scalar multiples?
If parallel, take an anchor of one line and test if it lies on the other → coincident (yes) or parallel only (no).
If not parallel, set vector equations equal (use λ and μ) and write three component equations.
Solve any two for λ and μ; substitute into the third.
If the third equation is satisfied → intersecting (compute the point). If not → skew.

Worked examples

WE 1

Show two lines are parallel

Show that the lines r₁ = (2, −3, 1) + λ(2, −4, 6) and r₂ = (5, 1, −2) + μ(1, −2, 3) are parallel.

Compare direction vectors b₁ = (2, −4, 6); b₂ = (1, −2, 3) Check if b₁ is a scalar multiple of b₂ b₁ = 2 × b₂ = 2(1, −2, 3) = (2, −4, 6) ✓ Lines are parallel we don’t yet know whether they’re coincident — would need a point check

WE 2

Show two lines are coincident

Show that the lines r₁ = (1, 2, −1) + s(2, 1, −3) and r₂ = (5, 4, −7) + t(−4, −2, 6) are coincident.

Step 1: Check direction vectors are scalar multiples b₂ = (−4, −2, 6) = −2 × (2, 1, −3) = −2 b₁ ✓ Step 2: Check anchor of l₂, (5, 4, −7), lies on l₁ x: 5 = 1 + 2s → s = 2 y: 4 = 2 + s → s = 2 ✓ z: −7 = −1 − 3s → s = 2 ✓ Same line → coincident parallel + shared point = coincident; without the point check it could just be parallel

WE 3

Show two lines intersect and find the intersection

Show that the lines r₁ = (3, 3, 0) + λ(1, 2, −1) and r₂ = (1, 9, −8) + μ(2, −1, 3) intersect, and find the point of intersection.

Step 1: Directions (1, 2, −1) and (2, −1, 3) — not scalar multiples → not parallel Step 2: Set r₁ = r₂ component-wise x: 3 + λ = 1 + 2μ → λ − 2μ = −2 … (1) y: 3 + 2λ = 9 − μ → 2λ + μ = 6 … (2) z: −λ = −8 + 3μ → λ + 3μ = 8 … (3) Step 3: Solve (1) and (2) From (1): λ = 2μ − 2; sub into (2): 5μ = 10 → μ = 2, λ = 2 Step 4: Check (3): 2 + 3(2) = 8 ✓ Step 5: Sub λ = 2 into r₁ r₁ = (3+2, 3+4, 0−2) = (5, 7, −2) Intersect at (5, 7, −2) always verify with the third equation — and double-check by substituting μ into r₂

WE 4

Show two lines are skew

Show that the lines r₁ = (1, 0, 2) + λ(2, 1, −1) and r₂ = (3, −2, 5) + μ(1, −1, 2) are skew.

Step 1: Directions (2, 1, −1) and (1, −1, 2) — not scalar multiples → not parallel Step 2: Set r₁ = r₂ x: 1 + 2λ = 3 + μ → 2λ − μ = 2 … (1) y: λ = −2 − μ → λ + μ = −2 … (2) z: 2 − λ = 5 + 2μ → λ + 2μ = −3 … (3) Step 3: Solve (1) + (2): 3λ = 0 → λ = 0, μ = −2 Step 4: Check (3): 0 + 2(−2) = −4 ≠ −3 ✗ Inconsistent → lines are skew not parallel and don’t intersect — only possible in 3D

WE 5

Distinguish parallel from coincident

Determine whether the lines r₁ = (2, 1, −3) + s(1, −2, 4) and r₂ = (5, 4, 1) + t(2, −4, 8) are parallel, coincident, or neither.

Step 1: Check directions b₂ = (2, −4, 8) = 2(1, −2, 4) = 2b₁ ✓ → parallel Step 2: Check if (5, 4, 1) lies on l₁ x: 5 = 2 + s → s = 3 y: 4 = 1 − 2s → s = −3/2 ✗ Different s values → (5, 4, 1) is NOT on l₁ Parallel but not coincident if the directions match but the anchor of one line isn’t on the other, the lines run side by side

WE 6

Find a value that makes two lines parallel

The lines r₁ = (2, −1, 5) + λ(3, k, 6) and r₂ = (1, 4, −2) + μ(2, −4, 4) are parallel. Find the value of k.

Step 1: For parallel, b₁ = c·b₂ for some scalar c (3, k, 6) = c(2, −4, 4) Step 2: Use the known components to find c x: 3 = 2c → c = 3/2 z: 6 = 4c → c = 3/2 ✓ (consistent) Step 3: Apply c to the y-component k = −4c = −4 × (3/2) = −6 k = −6 always verify c is the same from two known components before solving for the unknown

💡 Top tips

Use different parameters for the two lines (λ and μ) — same parameter would force them to be coupled.
Skew is impossible in 2D — only ever a 3D answer.
Always verify the third equation when solving the intersection system — that’s the test.
For parallel lines, the simplest check is whether one line’s anchor lies on the other.
Not all “scalar multiples” use positive scalars: (2, −4, 6) is parallel to (−1, 2, −3) via k = −2.

⚠ Common mistakes

Using the same parameter for both lines — must be λ and μ.
Stopping at “directions are scalar multiples” and calling the lines coincident — they could just be parallel.
Forgetting to check the third equation when finding intersection — without that check, you can’t distinguish intersect from skew.
Calling lines “intersecting” because the paths cross in 2D — in 3D, the lines may cross visually in a sketch but pass at different “heights”.
Sign errors when rearranging the three component equations into λ–μ standard form.

Next: Angle Between Two Lines. Once you know two lines aren’t parallel, the natural follow-up is “what angle do they make?” — found from the scalar product of their direction vectors. Same formula as the angle between two vectors: cosθ = (b₁ · b₂) / (|b₁| |b₂|).

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