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IGCSE / O Level Mathematics · Topic 7 Vectors and transformations · Chapter 31 · pp. 440–451

Vectors

A vector is a move: so far across, so far up or down. Two numbers in a column say it all. This chapter shows how to write a move, add and subtract moves, stretch them, measure how long they are, and use them to prove facts about shapes.

Learning objectives

  1. Describe a translation with a column vector, and name vectors with two letters (AB) or one letter (a).
  2. Add and subtract vectors, and show the sums as diagrams.
  3. Multiply a vector by a scalar and spot parallel vectors.
  4. Calculate the magnitude |a| of a vector and give position vectors from O.
  5. Use vectors to prove results about shapes, including midpoints and ratios.

Book exercises: 31.1 to 31.7 and Student assessments 1 to 3 (pp. 440–451).

Translations and column vectors

🔑 Key ideas

  • A translation slides every point the same distance in the same direction.
  • A column vector xy says: x across (right is +, left is −), then y up or down (up is +, down is −).
  • AB is the move from A to B. The arrow shows the direction, and the order of the letters matters.
  • A single letter such as a names a vector too. In handwriting, underline it: a.
  • The reverse move has the opposite signs: BA = −AB.
A B C D AB = (1, 3): 1 right, 3 up BC = (2, 0): 2 right, 0 up CD = (0, −2): 0 across, 2 down DA = (−3, −1): 3 left, 1 down
Each side of the shape is a vector. Add the four and you arrive back where you started: (1,3) + (2,0) + (0,−2) + (−3,−1) = (0,0).
The moveColumn vector
3 right, 2 up32
4 left, 1 up−41
5 right, no vertical move50
2 left, 6 down−2−6

💡 Across first, then up. Think of reading coordinates: (x, y) in a row becomes x on top of y in a column.

📘 Example 1 · Describe the move AB where A is (1, 2) and B is (4, 6)

  1. Across: from x = 1 to x = 4 is 4 − 1 = 3 to the right.
  2. Up: from y = 2 to y = 6 is 6 − 2 = 4 up.
  3. Write them in a column: AB = 34.
  4. Check: start at A (1, 2), move 3 right and 4 up and you land on (4, 6). ✔
a −a a = (2, 5) −a = (−2, −5) Same length, opposite direction. Change the sign of both numbers.
The negative of a vector points the opposite way. It is the move that undoes a.

📘 Example 2 · The book's shape: BC, CD and DA from the diagram above

  1. BC: B to C is 2 squares right and none up or down, so BC = 20.
  2. CD: C to D is none across and 2 squares down, so CD = 0−2.
  3. DA: D to A is 3 squares left and 1 down, so DA = −3−1.
  4. Check the reverse: AD = −DA = 31, which is 3 right and 1 up. ✔

📘 Example 3 · AB = (−2, 5). Write BA and say what the move is

  1. BA is the opposite of AB: change both signs.
  2. BA = 2−5.
  3. In words: 2 right and 5 down. AB was 2 left and 5 up, so they cancel. ✔

✏️ Try it

  1. A is (1, 2) and B is (4, 6). Write AB as a column vector.
  2. CD = (−2, 5). Write DC.
  3. Describe the translation (0, −4) in words.
Answers

1. (3, 4) · 2. (2, −5) · 3. 4 units down, no sideways move

Next: adding and subtracting →

Adding vectors

🔑 Key ideas

  • a + b means the move a followed by the move b. Draw b starting where a ends (nose to tail).
  • The single move from the very start to the very end is the resultant, a + b.
  • With numbers: add the tops, add the bottoms.
  • Order does not matter: a + b = b + a.
a b a + b a = (2, 5), b = (−3, −2) a + b = (2 + (−3), 5 + (−2)) = (−1, 3) b starts where a ends. The dashed arrow is the shortcut.
a followed by b is the same as the single move a + b.

Subtracting vectors

🔑 Key ideas

  • a − b = a + (−b): do a, then the opposite of b.
  • With numbers: subtract the tops, subtract the bottoms.
  • Careful: a − b ≠ b − a. They are opposites of each other.
a −b a − b a = (2, 5), b = (−3, −2) −b = (3, 2) a − b = (2 − (−3), 5 − (−2)) = (5, 7) Draw −b (b reversed) after a.
Subtracting b is the same as adding −b.

📘 Example 4 · a = (3, 4), b = (−2, 1), c = (−4, −3). Find a + b and b − c

  1. a + b: tops 3 + (−2) = 1, bottoms 4 + 1 = 5. So a + b = 15.
  2. b − c: tops −2 − (−4) = −2 + 4 = 2.
  3. Bottoms: 1 − (−3) = 1 + 3 = 4.
  4. So b − c = 24.
  5. Check: c + (b − c) = (−4 + 2, −3 + 4) = (−2, 1) = b. ✔

📘 Example 5 · Same vectors. Find a + c − b

  1. a + c: (3 + (−4), 4 + (−3)) = (−1, 1).
  2. Now subtract b: (−1 − (−2), 1 − 1) = (1, 0).
  3. So a + c − b = 10.

📘 Example 6 · AB = (3, 1) and BC = (−1, 4). Find AC

  1. Going A to B to C is the same as going straight from A to C.
  2. AC = AB + BC = (3 + (−1), 1 + 4).
  3. AC = 25.

✏️ Try it

  1. a = (1, 4) and b = (3, −2). Find a + b.
  2. Find a − b.
  3. Find b − a. What do you notice?
Answers

1. (4, 2) · 2. (1 − 3, 4 − (−2)) = (−2, 6) · 3. (2, −6): it is the negative of a − b, so b − a = −(a − b)

Next: scalar multiples →

Multiplying a vector by a scalar

🔑 Key ideas

  • A scalar is an ordinary number. Multiplying a vector by it multiplies both parts.
  • If the scalar is positive the direction is kept. If it is negative the direction is reversed.
  • If the scalar is bigger than 1 (ignoring the sign) the vector gets longer; between 0 and 1 it gets shorter.
  • Vectors that are multiples of each other are parallel.
a 2a −a ½a a = (1, 2) 2a = (2, 4) −a = (−1, −2) ½a = (½, 1) All parallel to a.
Multiples of a: twice as long, the same length reversed, and half as long.

📘 Example 7 · a = (2, −4). Express b = −a, c = 2a, d = ½a and e = −1½a as vectors

  1. b = −a: change both signs: −24.
  2. c = 2a: double both: 4−8.
  3. d = ½a: halve both: 1−2.
  4. e = −1½a: −1.5 × 2 = −3 and −1.5 × (−4) = 6, so −36.

📘 Example 8 · p = (1, −2) and q = (3, 4). Find 2p + q and ½(q − p)

  1. 2p = (2, −4). Add q: (2 + 3, −4 + 4) = (5, 0).
  2. q − p = (3 − 1, 4 − (−2)) = (2, 6).
  3. Halve it: ½(q − p) = (1, 3).

📘 Example 9 · Write (−6, 9) in terms of a = (−2, 3)

  1. Compare the tops: −6 ÷ (−2) = 3.
  2. Compare the bottoms: 9 ÷ 3 = 3. Same scalar, so it works.
  3. (−6, 9) = 3a. The two vectors are parallel.

✏️ Try it

  1. a = (2, −3). Find 3a.
  2. Find −½a.
  3. Write (−6, 9) in terms of a.
Answers

1. (6, −9) · 2. (−1, 1½) · 3. (−6, 9) = −3a, since −3 × 2 = −6 and −3 × (−3) = 9

Next: magnitude and position vectors →

The magnitude of a vector Extended

🔑 Key ideas

  • The magnitude is the length of the arrow. Write it |a| or |AB|.
  • The across part and the up part make the two short sides of a right-angled triangle. Use Pythagoras.
  • |(x, y)| = √(x² + y²). Squaring removes the minus signs, so a magnitude is never negative.
  • |ka| = |k| × |a|, but |a + b| is usually not |a| + |b|.
3 4 |a| a = (3, 4) |a|² = 3² + 4² = 9 + 16 = 25 |a| = √25 = 5 The 3-4-5 triangle again.
The magnitude of (3, 4) is the hypotenuse of a triangle with sides 3 and 4.

📘 Example 10 · Find |a| for a = (3, 4) and |BC| for BC = (−6, 8)

  1. |a| = √(3² + 4²) = √(9 + 16) = √25 = 5.
  2. |BC| = √((−6)² + 8²). Square −6 to get +36.
  3. = √(36 + 64) = √100 = 10.

📘 Example 11 · a = (4, −3), b = (−5, 7). Find |2a − b| to 1 d.p.

  1. 2a = (8, −6).
  2. 2a − b = (8 − (−5), −6 − 7) = (13, −13).
  3. |2a − b| = √(13² + (−13)²) = √(169 + 169) = √338.
  4. √338 = 18.38… so 18.4 to 1 d.p.
  5. Check: 18² = 324 and 19² = 361, and 338 is nearer 324. ✔

Position vectors Extended

🔑 Key ideas

  • A position vector is a vector from a fixed point, usually the origin O, to a point.
  • If A is the point (2, 6), then OA = (2, 6): the numbers are the same.
  • Vector between two points: AB = OB − OA (end minus start).
A (2, 6) O OA = (2, 6) 2 across and 6 up from O. If B is (5, 2): AB = OB − OA = (5 − 2, 2 − 6) = (3, −4)
A position vector has the same numbers as the point's coordinates.

📘 Example 12 · A is (−1, 2) and B is (3, −1). Find OA, OB and AB, and |AB|

  1. OA = (−1, 2) and OB = (3, −1).
  2. AB = OB − OA = (3 − (−1), −1 − 2) = (4, −3).
  3. |AB| = √(4² + (−3)²) = √25 = 5.

🧪 Try it: vector playground

a =

b =

scalar

a b a + b ka 1 square = 1 unit
Solid arrows: a and b from the origin. Dashed: a + b. Thin: ka. Each square is the size shown bottom left.

Working

    ✏️ Try it

    1. Find |(8, 6)|.
    2. Find |(−5, −12)|.
    3. Find |(2, 3)| to 1 d.p.
    Answers

    1. √(64 + 36) = √100 = 10 · 2. √(25 + 144) = √169 = 13 · 3. √13 = 3.605… = 3.6

    Next: vector geometry →

    Vector geometry Extended

    🔑 Key ideas

    • A vector is not fixed in place. Any arrow with the same length and direction is the same vector.
    • In a parallelogram ABCD, opposite sides give equal vectors: DC = AB and BC = AD.
    • Reversing the letters reverses the vector: BA = −AB.
    • To find a vector, go from the start to the end by any route of vectors you know. Along an arrow is +, against an arrow is −.
    • If two vectors are multiples of each other they are parallel. That is how parallel lines are proved.
    A B C D a b a + b AB = a, AD = b DC = AB = a BC = AD = b AC = a + b BD = −a + b
    In the parallelogram ABCD, the opposite sides are equal vectors and the diagonal AC is a + b.
    1. Pick a route from the start to the end 2. Write each leg with: + along, − against 3. Add and simplify collect a terms and b terms
    The method for every vector geometry question.

    Ratios along a line Extended

    APB 2 parts3 parts AP : PB = 2 : 3, so AP = ⅖ AB and PB = ⅗ AB Midpoint M: AM : MB = 1 : 1, so AM = ½ AB
    A ratio m : n cuts the line into m + n equal parts. The first piece is m/(m + n) of the whole.

    📘 Example 13 · ABCD is a parallelogram with AB = a and AD = b. M is the midpoint of AC. Find AC, BD and AM

    1. AC: go A to B to C. BC = AD = b, so AC = AB + BC = a + b.
    2. BD: go B to A to D: BA = −a, then AD = b. So BD = −a + b (or b − a).
    3. M is the midpoint, so AM = ½AC = ½(a + b).
    4. Check with numbers: a = (2, 0), b = (1, 3). AC = (3, 3) and AM = (1.5, 1.5) which is half of AC. ✔

    📘 Example 14 · Book question: T is the midpoint of PS, R divides QS in the ratio 1 : 3, PT = a and PQ = b. Show RT = ¼(2a − 3b)

    1. PS = 2PT = 2a, because T is the midpoint.
    2. QS = QP + PS = −b + 2a = 2a − b.
    3. QR is 1 part out of 4: QR = ¼QS = ¼(2a − b).
    4. PR = PQ + QR = b + ¼(2a − b) = ¼(4b + 2a − b) = ¼(2a + 3b).
    5. RT = RP + PT = −¼(2a + 3b) + a.
    6. = ¼(−2a − 3b + 4a) = ¼(2a − 3b). ✔ Shown.

    📘 Example 15 · LP = a, LQ = b, PM = 3LP and QN = 3LQ. Prove PQ is parallel to MN and MN = 4PQ

    1. LM = LP + PM = a + 3a = 4a. In the same way LN = 4b.
    2. PQ = PL + LQ = −a + b = b − a.
    3. MN = ML + LN = −4a + 4b = 4(b − a).
    4. So MN = 4PQ: they are multiples, hence parallel, and MN is four times as long. ✔

    ✏️ Try it

    1. ABCD is a parallelogram, AB = a, AD = b. Write DC.
    2. Write BC.
    3. Write BD.
    Answers

    1. DC = a (opposite side to AB) · 2. BC = b (opposite side to AD) · 3. BD = BA + AD = −a + b = b − a

    Next: summary and quiz →

    Chapter summary

    • A column vector xy is x across and y up. AB goes from A to B and BA = −AB.
    • Add and subtract the tops and the bottoms. a + b is a followed by b; a − b = a + (−b).
    • A scalar multiplies both parts. Multiples of a vector are parallel to it.
    • Magnitude: |(x, y)| = √(x² + y²). Never negative.
    • Position vector OA has the same numbers as the point A. AB = OB − OA.
    • In geometry: equal vectors are the same length and direction wherever they sit; find a route, add the legs, simplify.
    • Ratio m : n on a line: the first part is m/(m + n) of the whole.

    📄 Practice questions

    The book's Exercises 31.1 to 31.7 and Student assessments 1 to 3 are in the printable assignment for this chapter. Its figures are given there as coordinates or descriptions.

    Mini-quiz

    1. The column vector (2, −3) describes a translation of what?
    2. AB = (4, −1). What is BA?
    3. a = (2, 5) and b = (−3, −2). What is a + b?
    4. a = (2, 5) and b = (−3, −2). What is a − b?
    5. a = (−2, 4). What is 3a?
    6. Which vector is parallel to (2, −4)?
    7. What is the magnitude of the vector (3, 4)?
    8. What is the magnitude of (−5, 12)?
    9. AB = (3, 1) and BC = (−1, 4). What is AC?
    10. A point P is at (−3, 2) on a grid with origin O. What is the position vector OP?
    11. M is the midpoint of AB and AM = a. Which is AB?
    12. The vector a has magnitude 5. What is the magnitude of −2a?

    Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 31 (Vectors). Explanations, the Try it answers, the tool and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.