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S5 Maths: Vectors – Dot Product, Parallel and Perpendicular (Paper 2: Intermediate)
S5 Maths: Vectors – Dot Product, Parallel and Perpendicular (Paper 2: Intermediate)
Apply the dot product to parallel, perpendicular and section problems, working backwards from conditions, ten items with answers.
Direct answers This paper centres on the dot product and positional relations, with every value computed from components.
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: S5 Maths: Vectors |
| Level | Intermediate |
02 Questions (10 items)
- FoundationGiven a = (2, 5) and b = (3, -1), find a dot b.
- FoundationGiven c = (3, 4), find its unit vector.
- FoundationGiven u = (2, -1) parallel to v = (k, 4), find k.
- IntermediateGiven a = (1, 2) perpendicular to b = (k, 3), find k.
- IntermediateGiven d = (5, 12), find |d|.
- IntermediateGiven A(2, 1) and B(6, 7), find the midpoint of AB.
- ChallengeGiven A(0, 0) and B(8, 12), point P on AB with AP to PB as 1 to 3, find P.
- ChallengeFind the angle between a = (1, 0) and b = (1, root 3).
- ChallengeGiven a = (1, 2) and b = (3, 0), find |2a – b|.
- Error redoA student says a dot b = 0 must mean a or b is the zero vector. What is wrong?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 1 | Multiply matching components and add: 6 minus 5 is 1. |
| 2 | (0.6, 0.8) | Take the length 5 and divide each component by it. |
| 3 | k = -8 | Parallel means proportional components, so 2 times 4 equals negative 1 times k. |
| 4 | k = -6 | Perpendicular means a zero dot product, so k plus 6 is zero. |
| 5 | 13 | 25 plus 144 is 169, and its square root is 13. |
| 6 | (4, 4) | A midpoint averages the coordinates: 2 and 6 give 4, 1 and 7 give 4. |
| 7 | (2, 3) | P covers one quarter of AB, so 8 over 4 is 2 and 12 over 4 is 3. |
| 8 | 60 degrees | The dot product is 1 while the lengths are 1 and 2, so the cosine is 0.5 and the angle is 60 degrees. |
| 9 | About 4.12 (root 17) | 2a is (2, 4) and subtracting b gives (-1, 4), whose length is root 17, about 4.12. |
| 10 | Not necessarily; the vectors may simply be perpendicular | The zero vector gives a zero product, but perpendicular non-zero vectors also do. |
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.
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