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 1Show 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 2Show 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 3Show 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 4Show 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 5Distinguish 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 6Find 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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