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S6 Maths: Vectors and the Dot Product: Paper 1 (Basics)
S6 Maths: Vectors and the Dot Product: Paper 1 (Basics)
Ten items starting from vector length and the definition of the dot product, then addition, unit vectors and the perpendicular condition.
Direct answers Items 1-3: 5, 13, 0. Items 4-6: 45°, then (7, 6) with 9.2195, then (−4, 11). Items 7-9: (0.6, 0.8), −9, 25. Item 10: 23.
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 | Basics |
02 Questions (10 items)
- FoundationGiven a = (3, 4), find |a|.
- FoundationGiven a = (2, 5) and b = (4, 1), find a · b.
- FoundationGiven a = (6, −2) and b = (1, 3), find a · b and say whether the vectors are perpendicular.
- IntermediateGiven a = (1, 2) and b = (3, 1), find the angle between them.
- IntermediateGiven a = (2, −1) and b = (5, 7), find a + b and its length (4 d.p.).
- IntermediateGiven a = (1, 4) and b = (2, −1), find 2a − 3b.
- ChallengeFind the unit vector in the direction of (3, 4).
- ChallengeGiven a = (2, 3) is perpendicular to b = (k, 6), find k.
- ChallengeGiven a = (3, 4), find a · a and state how it relates to |a|.
- Error redoA student writes (2, 3) · (4, 5) = (8, 15). What is wrong and what is the correct answer?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | |a| = 5 | |a| = √(3² + 4²) = √25 = 5; a length is never negative. |
| 2 | a · b = 13 | Multiply matching components and add: 2 × 4 + 5 × 1 = 13. |
| 3 | a · b = 0, so they are perpendicular | 6 × 1 + (−2) × 3 = 0, and a zero dot product means a right angle. |
| 4 | 45° | a · b = 5 and the lengths are √5 and √10, so cos θ ≈ 0.7071 and θ = 45°. |
| 5 | a + b = (7, 6), length about 9.2195 | Adding components gives (7, 6); its length is √85 ≈ 9.2195. |
| 6 | (−4, 11) | 2a = (2, 8) and 3b = (6, −3), so the difference is (−4, 11). |
| 7 | (0.6, 0.8) | Divide by the length 5 to get (0.6, 0.8), which has length 1. |
| 8 | k = −9 | Perpendicular means 2k + 18 = 0, so k = −9. |
| 9 | a · a = 25 = |a|² | a · a = 9 + 16 = 25, and |a| = 5, so the dot product of a with itself is |a|². |
| 10 | A dot product gives a scalar; the answer is 23 | Multiply matching components and add: 8 + 15 = 23. The result is a number, not a vector. |
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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