IB Maths AI HL Vector Properties Paper 1 & 2 Scalar multiples ~7 min read

Parallel Vectors

Two vectors are parallel when one is a scalar multiple of the other: a = kb for some real k. A positive k means same direction; a negative k means opposite direction. Either way, every component of a is the same constant times the matching component of b, so the test is simple: divide corresponding components and see if you get the same ratio every time.

📘 What you need to know

Parallel definition: a is parallel to b iff there exists a scalar k with a = kb. Both vectors must be non-zero.
Component test: a₁b₁ = a₂b₂ = a₃b₃. If all three component-ratios are equal, that common ratio is the scalar k.
Sign of k: k > 0 → same direction (just scaled); k < 0 → opposite direction (reversed and scaled).
Magnitude relation: |a| = |k| |b|. The scalar’s absolute value gives the length ratio.
Factoring trick: factor out a common constant from each vector to spot a shared “direction vector” — if both reduce to the same direction vector, they’re parallel.
Zero components: a zero in one vector forces a zero in the matching slot of any parallel vector. (0/0 doesn’t give a ratio; just check it matches.)

The component-ratio test

If a = kb, then every component of a equals k times the matching component of b. Compute the three ratios a₁/b₁, a₂/b₂, a₃/b₃ — if they’re all equal, the vectors are parallel and the common value is your scalar k. If even one ratio disagrees, the vectors are not parallel. If one component is zero on both, skip that ratio (it carries no information); if it’s zero on one but not the other, the vectors are not parallel.

Five vectors, all parallel: three teal multiples of d (positive scalars, same direction) and two orange multiples (negative scalars, reversed direction). The dashed grey line is the common direction shared by all five — this is what “parallel” means.

Parallel vector test a = kb ⇔ a₁b₁ = a₂b₂ = a₃b₃ = k all three ratios must give the same value of k — one disagreement and the vectors are not parallel

Finding unknowns from a parallel condition

If a question tells you two vectors are parallel and one (or both) has a missing component, set up the equation a = kb. Pick any pair of corresponding components where both are known — that gives you k immediately. Then use this k to recover any unknown components in either vector by matching the remaining slots. This single technique solves every “find t such that the vectors are parallel” question on the syllabus.

Factoring shortcut: if both vectors contain a common factor, pull it out. a = 6i + 4j − 2k = 2(3i + 2j − k) and b = 9i + 6j − 3k = 3(3i + 2j − k) — both reduce to the same direction vector, so they’re parallel.

🧭 Recipe — test for parallel / find unknowns

Write both vectors in the same form (column or base-vector), so components line up.
Set up a = kb for an unknown scalar k.
Compute the ratio of one pair of matching components to find k.
Verify with the other ratios: if all three give the same k, vectors are parallel; if any differ, they’re not.
For unknown components: substitute the k from a known pair to find each unknown via a_i = k · b_i.

Worked examples

WE 1

Test for parallel and find the scalar

Determine whether the vectors a = (4, 2, −6)^T and b = (10, 5, −15)^T are parallel. If they are, state the scalar that maps a onto b.

compute the three component ratios b/a 10/4 = 5/2 5/2 = 5/2 −15/−6 = 15/6 = 5/2 all three equal ⇒ parallel parallel; b = (5/2)a, so k = 5/2 positive k ⇒ same direction; |b| = (5/2)|a|.

WE 2

Non-parallel case

Determine whether the vectors p = (3, −1, 4)^T and q = (6, −2, 7)^T are parallel.

compute the three ratios q/p 6/3 = 2 −2/−1 = 2 7/4 = 1.75 third ratio disagrees with the first two not parallel one mismatch is enough — the components don’t share a common scalar.

WE 3

Mixed forms — show parallel and find k

Show that the vectors a = (3, −9, 6)^T and b = i − 3j + 2k are parallel, and find the scalar k such that a = kb.

write b as a column b = (1, −3, 2)^T component ratios a / b 3/1 = 3 −9/−3 = 3 6/2 = 3 parallel; a = 3b, so k = 3 factor: a = (3, −9, 6) = 3(1, −3, 2) = 3b — confirms by inspection.

WE 4

Negative scalar — opposite direction

Show that the vectors a = 2i − 4j + 6k and b = −5i + 10j − 15k are parallel, and find the scalar that maps a onto b.

column form a = (2, −4, 6)^T b = (−5, 10, −15)^T ratios b / a −5/2 = −5/2 10/−4 = −5/2 −15/6 = −5/2 parallel; b = (−5/2)a, so k = −5/2 negative k means b points in the opposite direction to a, with |b| = (5/2)|a|.

WE 5

Find a missing component

The vectors u = (8, t, 12)^T and v = (2, −3, 3)^T are parallel. Find the value of t and state the scalar that maps v onto u.

set u = k v from first components: 8 = 2k → k = 4 check third: 12 = 3k = 3·4 = 12 ✓ find t using the middle component t = −3·k = −3·4 t = −12 t = −12; u = 4v always verify k with a second known pair before solving for the unknown.

WE 6

Two unknowns from a parallel condition

Find the values of a and b such that the vectors p = (a, 6, −4)^T and q = (3, 9, b)^T are parallel.

middle components both known → use them for k p = k q ⇒ 6 = 9k k = 6/9 = 2/3 first components: a = 3k a = 3·(2/3) = 2 third components: −4 = bk b = −4 / (2/3) = −4·(3/2) = −6 a = 2, b = −6 verify: p = (2, 6, −4) and q = (3, 9, −6); each ratio = 2/3 ✓

💡 Top tips

Use the pair with no unknowns first — that gives you k cleanly; then solve for any unknowns.
Compare ratios as fractions, not decimals — 5/2 and 2.5 look different but are equal; sticking to fractions avoids ambiguity.
For a negative ratio, the vectors point opposite ways — that still counts as parallel.
Factor out common multiples to spot parallelism quickly: 6i + 4j − 2k = 2(3i + 2j − k).
If one component is 0 in both, that pair gives no information — check the remaining two pairs only.

⚠ Common mistakes

Only checking two of three ratios — all three must agree before concluding “parallel”.
Sign errors with negative components: −9/−3 = +3, not −3. Two negatives make a positive.
Treating 0/0 as a valid ratio — skip that pair; check the others.
Concluding “not parallel” from a sign mismatch when both ratios are the same negative value — that’s still parallel (opposite direction).
Forgetting to convert when one vector is in column form and the other in base-vector form — line them up first.

Next up — Position & Displacement Vectors. The position vector of point A is just OA — the vector from the origin to A. The displacement vector AB from A to B is b − a, the difference of their position vectors. Knowing this single relationship unlocks every “find the vector from P to Q” question on the syllabus.

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