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Vectors I

14 Grade 10 Core Mathematics questions on Vectors I, with worked marking schemes. 14 questions are available on this topic in full.

  1. Question 1

    1 mark

    A surveyor records the displacement of a boundary peg as the column vector (−8, 6), in metres. The magnitude of this vector is

    A. 100 m

    B. 10 m

    C. 14 m

    D. 2 m

    (1 mark)

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  2. Question 2

    5 marks

    Three market stalls are located at A(−1, 2), B(3, 4) and C(7, 6) on a grid map. Show that the three stalls lie on a straight line.

    (5 marks)

    Marking scheme

    1. AB = OB − OA = (3 − (−1), 4 − 2) = (4, 2) — 1 mark. 2. BC = OC − OB = (7 − 3, 6 − 4) = (4, 2) — 1 mark. 3. AB = BC, so one vector is a scalar multiple of the other (k = 1) — 1 mark. 4. Hence AB is parallel to BC — 1 mark. 5. The two vectors share the common point B, therefore A, B and C are collinear — 1 mark.

  3. Question 3

    3 marks

    A vector is given as p = (−4, 6).

    (a) Find p.

    (b) Find −p.

    (c) State the relationship between p and −p.

    (3 marks)

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  4. Question 4

    3 marks

    A tractor leaves the store O and moves with displacement a = (4, 3) to a seedbed, then with displacement b = (2, −7) from the seedbed to a water point, the distances being in hundreds of metres.

    (a) Find, as a column vector, the displacement of the water point from the store.

    (b) A second tractor drives from the store straight to the water point. Write down its displacement vector and state how it is related to your answer in (a).

    (3 marks)

    Marking scheme

    (a) 1. Displacement from store = a + b = (4 + 2, 3 + (−7)) — 1 mark for method. 2. = (6, −4) — 1 mark. (b) 3. The second tractor's displacement is also (6, −4); it is equivalent to a + b, because a displacement depends only on the starting and finishing points and not on the route taken (triangle law of vector addition) — 1 mark.

  5. Question 5

    4 marks

    A translation T maps the point A(1, −4) onto A'(−5, 2).

    (a) State the translation vector T.

    (b) Find the coordinates of the image of B(3, 6) under the same translation.

    (4 marks)

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  6. Question 6

    4 marks

    On a grid map of a settlement scheme, the position vectors of a borehole A and a cattle dip B are OA = (−3, 2) and OB = (9, −3), in kilometres.

    (a) Find the column vector AB.

    (b) Find the magnitude of AB.

    (4 marks)

    Marking scheme

    (a) 1. AB = OB − OA — 1 mark for correct method. 2. AB = (9 − (−3), −3 − 2) = (12, −5) — 1 mark. (b) 3. |AB| = √(12² + (−5)²) = — 1 mark. 4. |AB| = 13 km — 1 mark (unit required).

  7. Question 7

    1 mark

    The displacement of a handcart on one trip is p = (2, −5). If the handcart makes three identical trips, the total displacement 3p is

    A. (6, −15)

    B. (6, −5)

    C. (6, 15)

    D. (5, −8)

    (1 mark)

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  8. Question 8

    1 mark

    Two displacements of a drone over a tea estate are a = (3, −2) and b = (−5, 4). The column vector a + b is

    A. (−2, 2)

    B. (−15, −8)

    C. (−2, −6)

    D. (8, −6)

    (1 mark)

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  9. Question 9

    1 mark

    On a map grid, a water tank is at A(2, 7) and a dispensary is at B(−3, 4). The column vector AB is

    A. (−5, 3)

    B. (−5, −3)

    C. (5, 3)

    D. (−1, 11)

    (1 mark)

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  10. Question 10

    1 mark

    A Grade 10 class lists quantities recorded during a visit to a school farm in Kirinyaga. Which one of the quantities listed below is a vector quantity?

    A. The mass of a bag of maize harvested from the plot

    B. The area of the land under French beans

    C. The displacement of a wheelbarrow from the store to the seedbed

    D. The temperature inside the greenhouse at noon

    (1 mark)

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  11. Question 11

    1 mark

    A footpath runs straight from P(−4, 3) to Q(6, −9) on a grid map of a market centre. The coordinates of the midpoint of PQ are

    A. (1, −6)

    B. (1, −3)

    C. (5, −6)

    D. (2, −6)

    (1 mark)

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  12. Question 12

    3 marks

    Distinguish between a scalar quantity and a vector quantity, and give one example of each from a farm in Kenya.

    (3 marks)

    Marking scheme

    1. Scalar: a quantity that has magnitude (size) only — 1 mark. 2. Vector: a quantity that has both magnitude and direction — 1 mark. 3. One acceptable example of each, e.g. scalar — mass of a 90 kg bag of maize; vector — displacement of a tractor 200 m due east across the plot — 1 mark (both examples needed for the mark).

  13. Question 13

    1 mark

    A translation maps the point R(3, −2) onto R'(−1, 5). The translation vector is

    A. (−4, 7)

    B. (4, −7)

    C. (2, 3)

    D. (−4, 3)

    (1 mark)

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  14. Question 14

    3 marks

    Given the column vectors a = (4, −3) and b = (−1, 5), find .

    (3 marks)

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