Vectors

50 maths questions with worked answers with answers for 3.4.3 Vectors in AQA GCSE Maths (spec 8300). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official AQA material.

1. Question 1 · Translation · G24

   What translation is represented by the vector 3−2?

   Answer:

   The top component 3 means 3 units right.
   The bottom component −2 means 2 units down.

2. Question 2 · Translation · G24

   Write the vector for 4 units left and 5 units up.

   Answer:

   4 units left gives horizontal component −4.
   5 units up gives vertical component 5.
   The vector is −45.

3. Question 3 · Translation · G24

   Translate (2, 1) by 3−4.

   Answer:

   Add horizontal components: 2 + 3 = 5.
   Add vertical components: 1 − 4 = −3.
   The image is (5, −3).

4. Question 4 · Translation · G24

   Translate (−1, 6) by −23.

   Answer:

   Add horizontal components: −1 − 2 = −3.
   Add vertical components: 6 + 3 = 9.
   The image is (−3, 9).

5. Question 5 · Translation · G24

   Find the vector that maps (1, 2) to (5, −1).

   Answer:

   Horizontal change = 5 − 1 = 4.
   Vertical change = −1 − 2 = −3.
   The vector is 4−3.

6. Question 6 · Translation · G24

   Find the image of (0, −3) under translation −52.

   Answer:

   Horizontal coordinate: 0 − 5 = −5.
   Vertical coordinate: −3 + 2 = −1.
   The image is (−5, −1).

7. Question 7 · Translation · G24

   A point moves from (−3, 4) to (2, 4). Give the translation vector.

   Answer:

   Horizontal change = 2 − (−3) = 5.
   Vertical change = 4 − 4 = 0.
   The vector is 50.

8. Question 8 · Translation · G24

   Find the translation taking (7, 2) to (−1, −2).

   Answer:

   Horizontal change = −1 − 7 = −8.
   Vertical change = −2 − 2 = −4.
   The vector is −8−4.

9. Question 9 · Translation · G24

   A point is translated by 06. Describe its movement.

   Answer:

   The horizontal component is 0, so there is no horizontal movement.
   The vertical component is 6, so move 6 units up.

10. Question 10 · Translation · G24

    A point at (3, −2) is translated to (0, 5). Give the vector.

    Answer:

    Horizontal change = 0 − 3 = −3.
    Vertical change = 5 − (−2) = 7.
    The vector is −37.

11. Question 11 · Components · G25

    If a = 25 and b = 3−1, find a + b.

    Answer:

    Add horizontal components: 2 + 3 = 5.
    Add vertical components: 5 + (−1) = 4.
    a + b = 54.

12. Question 12 · Components · G25

    If a = 62 and b = 15, find a − b.

    Answer:

    Subtract corresponding components: 6 − 1 = 5 and 2 − 5 = −3.
    a − b = 5−3.

13. Question 13 · Components · G25

    If a = −24, find 3a.

    Answer:

    Multiply both components by 3: 3 × (−2) = −6 and 3 × 4 = 12.
    3a = −612.

14. Question 14 · Components · G25

    If b = 8−6, find b2.

    Answer:

    Divide both components by 2: 8 ÷ 2 = 4 and −6 ÷ 2 = −3.
    b2 = 4−3.

15. Question 15 · Components · G25

    If a = 37, find −a.

    Answer:

    Multiply each component by −1.
    −a = −3−7.

16. Question 16 · Components · G25

    Calculate 4−2 + −18.

    Answer:

    Add corresponding components: 4 + (−1) = 3 and −2 + 8 = 6.
    The result is 36.

17. Question 17 · Components · G25

    Calculate −34 − 2−5.

    Answer:

    Subtract corresponding components: −3 − 2 = −5 and 4 − (−5) = 9.
    The result is −59.

18. Question 18 · Components · G25

    Calculate 21−3 + 45.

    Answer:

    Double the first vector to get 2−6.
    Add components: 2 + 4 = 6 and −6 + 5 = −1.
    The result is 6−1.

19. Question 19 · Components · G25

    Calculate 321 − 214.

    Answer:

    Multiply components: 321 = 63 and 214 = 28.
    Subtract to get 4−5.

20. Question 20 · Components · G25

    Find x if x3 + 2−1 = 72.

    Answer:

    Equate horizontal components: x + 2 = 7.
    Subtract 2: x = 5.

21. Question 21 · Components · G25

    Find y if 4y − 12 = 36.

    Answer:

    Equate vertical components: y − 2 = 6.
    Add 2: y = 8.

22. Question 22 · Components · G25

    Find a if 2a = 10−4.

    Answer:

    Divide each component of 2a by 2: 10 ÷ 2 = 5 and −4 ÷ 2 = −2.
    a = 5−2.

23. Question 23 · Components · G25

    Which vector reverses 5−7?

    Answer:

    Reverse a vector by multiplying each component by −1.
    The reverse is −57.

24. Question 24 · Components · G25

    Find the magnitude of 34.

    Answer:

    Use Pythagoras: magnitude² = 3² + 4² = 25.
    Magnitude = √25 = 5.

25. Question 25 · Components · G25

    Find the magnitude of −512.

    Answer:

    Use Pythagoras: magnitude² = (−5)² + 12² = 169.
    Magnitude = √169 = 13.

26. Question 26 · Position vectors · G25

    A has position vector 23 and B has position vector 79. Find vector AB.

    Answer:

    Vector AB = position vector of B − position vector of A.
    Subtract components: 7 − 2 = 5 and 9 − 3 = 6.
    Vector AB = 56.

27. Question 27 · Position vectors · G25

    A = (−1, 4) and B = (3, −2). Find vector AB.

    Answer:

    Vector AB = position vector of B − position vector of A.
    Components: 3 − (−1) = 4 and −2 − 4 = −6.
    Vector AB = 4−6.

28. Question 28 · Position vectors · G25

    A = (4, 5) and B = (−2, 1). Find vector BA.

    Answer:

    Vector BA = position vector of A − position vector of B.
    Components: 4 − (−2) = 6 and 5 − 1 = 4.
    Vector BA = 64.

29. Question 29 · Position vectors · G25

    If vector OA = 3−1 and vector AB = 24, find vector OB.

    Answer:

    Follow the path from O to A to B: vector OB = vector OA + vector AB.
    Components: 3 + 2 = 5 and −1 + 4 = 3.
    Vector OB = 53.

30. Question 30 · Position vectors · G25

    If vector OA = 82 and vector AB = −35, find vector OB.

    Answer:

    Follow the path from O to A to B: vector OB = vector OA + vector AB.
    Components: 8 − 3 = 5 and 2 + 5 = 7.
    Vector OB = 57.

31. Question 31 · Position vectors · G25

    If vector OB = 96 and vector AB = 42, find vector OA.

    Answer:

    Vector OA = vector OB − vector AB.
    Components: 9 − 4 = 5 and 6 − 2 = 4.
    Vector OA = 54.

32. Question 32 · Position vectors · G25

    Find the midpoint of A(2, 6) and B(8, 10).

    Answer:

    Average the x-coordinates: (2 + 8) ÷ 2 = 5.
    Average the y-coordinates: (6 + 10) ÷ 2 = 8.
    Midpoint = (5, 8).

33. Question 33 · Position vectors · G25

    Find the midpoint of A(−4, 2) and B(2, −6).

    Answer:

    Average the x-coordinates: (−4 + 2) ÷ 2 = −1.
    Average the y-coordinates: (2 − 6) ÷ 2 = −2.
    Midpoint = (−1, −2).

34. Question 34 · Position vectors · G25

    P is halfway from A to B. If vector AB = 86, find vector AP.

    Answer:

    Halfway means vector AP is half of vector AB.
    8 ÷ 2 = 4 and 6 ÷ 2 = 3, so vector AP = 43.

35. Question 35 · Position vectors · G25

    If vector AB = 3−4, find vector BA.

    Answer:

    The reverse vector is found by changing both component signs.
    Vector BA = −34.

36. Question 36 · Paths · G25

    If vector AB = a and vector BC = b, express vector AC.

    Answer:

    Follow the path A to B to C.
    Vector AC = vector AB + vector BC = a + b.

37. Question 37 · Paths · G25

    If vector AB = a and vector AC = b, express vector BC.

    Answer:

    Follow B to A to C: vector BC = vector BA + vector AC.
    Vector BA = −a, so vector BC = −a + b = b − a.

38. Question 38 · Paths · G25

    If vector OA = a and vector OB = b, express vector AB.

    Answer:

    Travel from A to O to B: vector AB = vector AO + vector OB.
    Vector AO = −a, so vector AB = b − a.

39. Question 39 · Paths · G25

    If vector OA = a and vector AB = 2b, express vector OB.

    Answer:

    Follow O to A to B: vector OB = vector OA + vector AB.
    Therefore vector OB = a + 2b.

40. Question 40 · Paths · G25

    If vector PQ = 2a and vector QR = −a + b, express vector PR.

    Answer:

    Follow P to Q to R: vector PR = vector PQ + vector QR.
    2a + (−a + b) = a + b.

41. Question 41 · Paths · G25

    If vector AB = a and vector BC = 2a, express vector AC.

    Answer:

    Follow A to B to C: vector AC = vector AB + vector BC.
    a + 2a = 3a.

42. Question 42 · Paths · G25

    If vector AB = 3a and vector AC = a, express vector CB.

    Answer:

    Follow C to A to B: vector CB = vector CA + vector AB.
    Vector CA = −a, so vector CB = −a + 3a = 2a.

43. Question 43 · Paths · G25

    If vector AB = 41 and vector BC = −23, find vector AC.

    Answer:

    Follow A to B to C: vector AC = vector AB + vector BC.
    Add components: 4 + (−2) = 2 and 1 + 3 = 4.
    Vector AC = 24.

44. Question 44 · Paths · G25

    If vector PQ = 64 and vector PR = 1−2, find vector QR.

    Answer:

    Vector QR = vector PR − vector PQ.
    Components: 1 − 6 = −5 and −2 − 4 = −6.
    Vector QR = −5−6.

45. Question 45 · Parallel · G25 · Higher only

    Are vectors 23 and 69 parallel? Explain.

    Answer:

    Multiply both components of 23 by 3 to get 69.
    One vector is a scalar multiple of the other, so they are parallel.

46. Question 46 · Parallel · G25 · Higher only

    Are vectors 2−1 and −42 parallel? Explain.

    Answer:

    Multiply both components of 2−1 by −2 to get −42.
    They are parallel and point in opposite directions.

47. Question 47 · Parallel · G25 · Higher only

    Find k so 36 and k10 are parallel.

    Answer:

    The scale factor from 6 to 10 is 106 = 53.
    Apply it to the top component: k = 3 × 53 = 5.

48. Question 48 · Geometric proof · G25 · Higher only

    Vector OA = a and vector OB = b. M is the midpoint of segment AB. Express vector OM.

    Answer:

    Vector OM = vector OA + half of vector AB.
    Vector AB = b − a, so vector OM = a + b − a2.
    Simplify: vector OM = a + b2.

49. Question 49 · Geometric proof · G25 · Higher only

    If vector AB = a and vector CD = 2a, what can you conclude about these directed segments?

    Answer:

    Vector CD is the positive scalar multiple 2a of vector AB.
    For nonzero a, the segments are parallel in the same direction and CD is twice as long.

50. Question 50 · Geometric proof · G25 · Higher only

    If vector OP = a, vector OQ = 3a and a is nonzero, what can you conclude about O, P and Q?

    Answer:

    Vector OQ = 3 times vector OP and a is nonzero.
    The points O, P and Q are collinear; P is one-third of the way from O to Q.

More AQA GCSE Maths topics

• 3.1.1 Structure and calculation
• 3.1.2 Fractions, decimals and percentages
• 3.1.3 Measures and accuracy
• 3.2.1 Notation, vocabulary and manipulation
• 3.2.2 Graphs
• 3.2.3 Solving equations and inequalities
• 3.2.4 Sequences
• 3.3 Ratio, proportion and rates of change
• 3.4.1 Properties and constructions
• 3.4.2 Mensuration and calculation
• 3.4.3 Vectors
• 3.5 Probability
• 3.6 Statistics

Other Maths courses: Edexcel GCSE Maths, OCR GCSE Maths.