PassNova has 9 free A-level Maths practice questions on Vectors, each with a clear explanation. Practise them in the browser with instant feedback — 100% free, no sign-up, on any device. Updated for 2026.
Vectors: example questions & answers
9 worked examples with answers and explanations below. Practise them in the browser with instant feedback on every answer.
Find the magnitude of the vector 5i − 12j.
- A17
- B7
- C13✓
- D169
Answer: The magnitude is √(5² + (−12)²) = √(25 + 144) = √169 = 13. (Option A adds the components, D forgets to take the square root.)
Find the unit vector in the direction of 6i + 8j.
- A0.6i + 0.8j✓
- B3i + 4j
- C6i + 8j
- D0.6i − 0.8j
Answer: The magnitude is √(6² + 8²) = √100 = 10, so the unit vector is (6i + 8j)/10 = 0.6i + 0.8j. (Dividing by 2 instead of 10 gives B.)
Given a = 2i + 3j and b = 4i − j, find 3a − 2b.
- A14i + 7j
- B−2i + 7j
- C−2i − 11j
- D−2i + 11j✓
Answer: 3a = 6i + 9j and 2b = 8i − 2j, so 3a − 2b = (6 − 8)i + (9 − (−2))j = −2i + 11j. (Mishandling the sign on −2b's j-term gives the +7j distractor.)
The points A and B have position vectors a = 3i − 2j and b = 7i + 5j. Find the vector AB.
- A−4i − 7j
- B4i + 7j✓
- C4i + 3j
- D10i + 3j
Answer: AB = b − a = (7 − 3)i + (5 − (−2))j = 4i + 7j. (Option A is a − b; option C forgets the double negative in 5 − (−2).)
Find the distance between the points A(1, 2) and B(4, 6).
- A5✓
- B7
- C√7
- D25
Answer: Distance = √((4 − 1)² + (6 − 2)²) = √(9 + 16) = √25 = 5. (Option B adds 3 + 4; option D forgets the square root.)
Which of the following vectors is parallel to 3i − 4j?
- A3i + 4j
- B6i − 8j✓
- C4i − 3j
- D6i + 8j
Answer: A parallel vector is a scalar multiple: 6i − 8j = 2(3i − 4j). The others are not scalar multiples (e.g. 4i − 3j swaps the components).
The points A and B have position vectors 2i + 5j and 8i − 3j. Find the position vector of the midpoint M of AB.
- A10i + 2j
- B3i + 4j
- C5i + j✓
- D6i − 4j
Answer: The midpoint is (a + b)/2 = ((2 + 8)/2)i + ((5 + (−3))/2)j = 5i + j. (Option A is the sum without halving; option B is half of AB.)
Find the magnitude of the vector 2i − 3j + 6k.
- A11
- B√13
- C49
- D7✓
Answer: The magnitude is √(2² + (−3)² + 6²) = √(4 + 9 + 36) = √49 = 7. (Option A adds the components; B ignores the k-term.)
The points A and B have coordinates A(2, 1) and B(8, 13). The point P lies on AB such that AP : PB = 1 : 2. Find the coordinates of P.
- A(10, 14)
- B(4, 5)✓
- C(5, 7)
- D(6, 9)
Answer: AB = (8 − 2)i + (13 − 1)j = 6i + 12j. Since AP : PB = 1 : 2, AP = (1/3)AB = 2i + 4j, so P = A + AP = (2 + 2, 1 + 4) = (4, 5). (Option C is the midpoint; D reverses the ratio.)