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S6 Maths: Vectors and the Dot Product: Paper 2 (Intermediate)
S6 Maths: Vectors and the Dot Product: Paper 2 (Intermediate)
This paper focuses on the angle formula, scalar projection and expanding |a + b|, plus work done.
Direct answers Items 1-3: 45°, then 0 with 90°, then 3. Items 4-6: 6, 60°, 5. Items 7-9: 0, 4, 50 J. Item 10: about 9.899 against 10.
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 | Intermediate |
02 Questions (10 items)
- FoundationGiven a = (2, 1) and b = (1, 3), find the angle between them.
- FoundationGiven a = (−3, 4) and b = (4, 3), find a · b and the angle.
- FoundationGiven a = (3, 4), find its scalar projection on the x-axis direction.
- IntermediateGiven |a| = 3 and |b| = 4 with an angle of 60°, find a · b.
- IntermediateGiven a · b = 10, |a| = 5 and |b| = 4, find the angle.
- IntermediateGiven |a| = 3 and |b| = 4 with a perpendicular to b, find |a + b|.
- ChallengeGiven a = (1, 2) and b = (2, −1), find (a + b) · (a − b).
- ChallengeGiven a = (k, 2) is perpendicular to b = (3, −6), find k.
- ChallengeA force F = (6, 8) N moves an object through d = (3, 4) m. Find the work done.
- Error redoA student claims |a + b| always equals |a| + |b|. Test with a = (3, 4) and b = (4, 3).
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 45° | a · b = 5 with lengths √5 and √10 gives cos θ ≈ 0.7071, so θ = 45°. |
| 2 | a · b = 0, angle 90° | (−3) × 4 + 4 × 3 = 0, so the vectors are perpendicular and the angle is 90°. |
| 3 | 3 | Projection is a · b ÷ |b| with b = (1, 0), giving 3, the x-component of a. |
| 4 | a · b = 6 | a · b = |a||b| cos θ = 3 × 4 × 0.5 = 6. |
| 5 | 60° | cos θ = 10 ÷ 20 = 0.5, so the angle is 60°. |
| 6 | 5 | With a zero dot product |a + b|² = 9 + 16 = 25, so the length is 5. |
| 7 | 0 | (a + b) · (a − b) = |a|² − |b|² = 5 − 5 = 0, so the two are perpendicular. |
| 8 | k = 4 | 3k − 12 = 0 gives k = 4, and 12 − 12 = 0 confirms it. |
| 9 | 50 J | Work is F · d = 18 + 32 = 50 joules. |
| 10 | Not always; here |a + b| ≈ 9.899 while |a| + |b| = 10 | The sum has length √98 ≈ 9.899; equality holds only when the vectors point the same way. |
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 →
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