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S5 Maths: Vectors – Components and Length (Paper 1: Basics)
S5 Maths: Vectors – Components and Length (Paper 1: Basics)
Start from component addition, scalar multiplication and length to build the basic vector habits, ten items with answers.
Direct answers This paper focuses on component work and length, and every answer comes straight from the 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 | Foundation |
02 Questions (10 items)
- FoundationGiven a = (3, 4), find |a|.
- FoundationGiven a = (2, 1) and b = (5, 7), find a + b.
- FoundationGiven a = (6, -4), find 2a.
- IntermediateGiven a = (1, 2) and b = (4, -1), find a – b.
- IntermediateFind the unit vector in the direction of (3, 4).
- IntermediateGiven A(1, 2) and B(4, 6), find vector AB and its length.
- ChallengeGiven a = (2, -3) and b = (-1, 5), find 3a – 2b.
- ChallengeGiven a = (4, 3) and b = (3, -4), find a dot b and describe the relationship.
- ChallengeGiven a = (1, 2) and b = (4, 8), is b a multiple of a, and by what factor?
- Error redoA student says |a + b| always equals |a| + |b|. What is wrong, and what is a counterexample?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 5 | The length is the square root of 9 plus 16, which is 5. |
| 2 | (7, 8) | Vector addition adds matching components: 2 plus 5 is 7 and 1 plus 7 is 8. |
| 3 | (12, -8) | Scalar multiplication doubles each component. |
| 4 | (-3, 3) | Subtraction adds the negative vector: 1 minus 4 is negative 3 and 2 minus negative 1 is 3. |
| 5 | (3/5, 4/5) | Divide each component by the length, which is 5. |
| 6 | (3, 4), length 5 | AB is B minus A, and the length from components 3 and 4 is 5. |
| 7 | (8, -19) | 3a is (6, -9) and 2b is (-2, 10), so the difference is (8, -19). |
| 8 | a dot b = 0, so the vectors are perpendicular | The dot product is 12 minus 12, which is zero, and a zero dot product means the vectors are perpendicular. |
| 9 | Yes, b = 4a | Both components are multiplied by 4, so the vectors are parallel. |
| 10 | Not always; with a = (3, 0) and b = (0, 4), |a + b| = 5 but |a| + |b| = 7 | Equality holds only for the same direction; otherwise the triangle inequality applies. |
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.
Write the component working for each item, not just the answer.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.