S6 Maths: Vectors And The Dot Product: Paper 4 (Error Clinic)

S6 Maths: Vectors and the Dot Product: Paper 4 (Error clinic)

2026-11-11

S6 Maths: Vectors and the Dot Product: Paper 4 (Error clinic)

Ten items on the usual slips: treating a dot product as a vector, negative lengths, sign handling, confusing parallel with perpendicular, and dividing by a sum of components.

Direct answers　Items 1-3: 23, 10, 11. Items 4-6: 0 (perpendicular), parallel with a · b = 26, then √2 against 2. Items 7-9: −1, 180°, about 108.4°. Item 10: (0.6, 0.8).

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	20
Difficulty mix	Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1
Chapter	Chapter: S6 Maths: Vectors and the Dot Product
Level	Error clinic

02　Questions (10 items)

FoundationEvaluate (2, 3) · (4, 5) and explain why the answer is not a vector.
FoundationFind the length of (−6, 8).
FoundationEvaluate (7, −2) · (3, 5) and name a common sign error.
IntermediateDecide whether a = (4, −3) and b = (9, 12) are perpendicular.
IntermediateGiven a = (2, 3) and b = (4, 6), are they parallel or perpendicular?
IntermediateCompare |a + b| with |a| + |b| for a = (1, 0) and b = (0, 1).
ChallengeGiven a = (8, 2) is perpendicular to b = (k, 4), find k.
ChallengeFind the angle between a = (1, 2) and b = (−2, −4), and explain.
ChallengeGiven a = (1, 1) and b = (−2, 1), find the angle (1 d.p.).
Error redoA student writes the unit vector of (3, 4) as (3/7, 4/7). What is wrong?

03　Answers and one-line notes

#	Answer	One-line note
1	23, a scalar	2 × 4 + 3 × 5 = 23; the dot product multiplies matching components and adds, leaving a number.
2	10	√(36 + 64) = 10; simply adding the components to get 2 is a common slip.
3	11	21 − 10 = 11; writing −2 × 5 as +10 wrongly gives 31.
4	Yes, since a · b = 0	36 − 36 = 0 makes them perpendicular; their lengths of 5 and 15 play no part.
5	Parallel, with a · b = 26	b = 2a makes them parallel; the dot product 26 is not zero, so they are not perpendicular.
6	√2 ≈ 1.414 against 2	The sum has length √2 ≈ 1.4142 while the lengths add to 2; equality needs the same direction.
7	k = −1	8k + 8 = 0 gives k = −1, and −8 + 8 = 0 checks it.
8	180°	With cos θ = −10 ÷ 10 = −1 the angle is 180°, so the vectors point in opposite directions.
9	about 108.4°	a · b = −1 gives cos θ ≈ −0.3162, so θ ≈ 108.4°; the negative sign marks an obtuse angle.
10	The divisor is the length 5; the answer is (0.6, 0.8)	Divide by the length 5, not by 3 + 4, to get (0.6, 0.8).

04　What to do after marking

After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
Repeat the same item two days later; move to the next paper only when both attempts are correct.
Parents only need the answer and note columns; there is no need to explain every item.

Work out the dot product and both lengths first, then substitute.Tutor Registration →

Questions and answers are written originally by the TutorZone team and are not taken from any school, publisher or examination authority paper. Every value is checked by computer.