IGCSEMathsCambridge (CIE)International MathsExtendedRevision NotesVectors & TransformationsVectorsIntroduction to Vectors

Introduction to Vectors (Cambridge (CIE) IGCSE International Maths)

Revision Note

Jamie Wood

Expertise

Maths

Cambridge (CIE) International Maths: ExtendedEdexcel Maths A (Modular): Higher Unit 2

Basic Vectors

What is a vector?

  • Vectors represent a movement of a certain distance (magnitude) in a given direction

Vector arrow with the direction and magnitude labelled.

  • In one dimension, the sign of a number represents its direction 

    • For example, two objects with velocities 7 m/s and ‑7 m/s are travelling:

      • at the same speed (magnitude)

      • but in opposite directions

  • In two dimensions, vectors consist of x- and y-components

    • These show movement parallel to the x-  and y-axes

    • Components can be positive or negative 

How do I write vectors?

  • You can write the two components of a vector as a column vector

    • E.g. open parentheses table row 3 row 2 end table close parentheses is 3 right and 2 up

Column vector notation (x y) indicating movement in the x and y directions.

  • A lower-case letter can be used to represent the vector if the two components are not known

    • Exams use bold letters

      • a, b, ...

    • You should write underlined letters

      • a, b, ...

  • Vectors can also be described by their start and end points

    • If points (A , B , ...) are given

      • stack A B with rightwards arrow on top means the vector from A  to B

      • The order of the letters matters

Vector between two points A and B.

How are vectors used on a grid?

  • Vectors are often drawn on a grid (with or without x- and y-axes)

    • They can also be represented using column vectors

Three vectors a, b and c shown on a grid.

bold a equals open parentheses table row 3 row 4 end table close parentheses comma space space bold b equals open parentheses table row 2 row cell negative 4 end cell end table close parentheses comma space space bold c equals open parentheses table row 2 row 0 end table close parentheses

How do I multiply a vector by a scalar?

  • A scalar is a number with a magnitude but no direction

  • When a vector is multiplied by a positive scalar:

    • the magnitude of the vector changes

    • its direction stays the same

  • For a vector represented as a column vector

    • each of the numbers in the column vector will be multiplied by the scalar

Diagram showing vectors a, 1/2 a and 2a.

bold a equals open parentheses table row 4 row cell negative 2 end cell end table close parentheses comma space space space space space 2 bold a equals open parentheses table row cell 2 cross times 4 end cell row cell 2 cross times negative 2 end cell end table close parentheses equals open parentheses table row 8 row cell negative 4 end cell end table close parentheses comma space space space space 1 half bold a equals open parentheses table row cell 0.5 cross times 4 end cell row cell 0.5 cross times negative 2 end cell end table close parentheses equals open parentheses table row 2 row cell negative 1 end cell end table close parentheses

  •  When a vector is multiplied by a negative scalar:

    • the magnitude of the vector changes

    • its direction reverses

Diagram showing vectors a, -a and -2a.

bold a equals open parentheses table row 4 row cell negative 2 end cell end table close parentheses comma space space space space space minus bold a equals open parentheses table row cell negative 4 end cell row 2 end table close parentheses comma space space space space minus 2 bold a equals open parentheses table row cell negative 8 end cell row 4 end table close parentheses

How do I add vectors?

  • To add vectors you add their components

    • For column vectors, add the tops together and bottoms together

      • open parentheses 2 over 1 close parentheses plus open parentheses 1 fourth close parentheses equals blank open parentheses 3 over 5 close parentheses

  • Visually, the vector a + b is the shortest route

    • from the start of a

    • to the end of b

Diagram showing the addition of vectors a and b.

How do I subtract vectors?

  • To subtract vectors you subtract their components

    • open parentheses 2 over 1 close parentheses minus open parentheses 1 fourth close parentheses equals blank open parentheses fraction numerator 1 over denominator negative 3 end fraction close parentheses

  • Subtracting a vector can also be thought of as adding a negative vector

    • open parentheses 2 over 1 close parentheses minus open parentheses 1 fourth close parentheses equals stretchy left parenthesis 2 over 1 stretchy right parenthesis plus stretchy left parenthesis fraction numerator negative 1 over denominator negative 4 end fraction stretchy right parenthesis equals blank open parentheses fraction numerator 1 over denominator negative 3 end fraction close parentheses

  • Visually, the vector a - b is the shortest route

    • from the start of a

    • to the end of -b

Diagram showing the subtraction of vector b from vector a.

Worked Example

The points A, B and C are shown on the following coordinate grid.

Three points on a Cartesian grid

a) Write the vectors stack A B with rightwards arrow on top comma space stack A C with rightwards arrow on top and stack C B with rightwards arrow on top as column vectors.

Start by drawing the three vectors onto the grid

Diagram showing points A(-4, 2), B(2, 4) and C(3, -4) on a grid. The vectors AB, AC and CB are drawn as arrows between the relevant pairs of points.

From A to B, it is 6 to the right and 2 up

  

From A to C, it is 7 to the right and 6 down

  

From C to B, it is 1 to the left and 8 up

  

b) Using the column vectors from a), confirm that stack A B with rightwards arrow on top minus stack A C with rightwards arrow on top equals stack C B with rightwards arrow on top.

Perform the subtraction on the column vectors

  stack A B with rightwards arrow on top minus stack A C with rightwards arrow on top equals open parentheses table row 6 row 2 end table close parentheses minus open parentheses table row 7 row cell negative 6 end cell end table close parentheses equals open parentheses table row cell 6 minus 7 end cell row cell 2 minus open parentheses negative 6 close parentheses end cell end table close parentheses equals open parentheses table row cell negative 1 end cell row 8 end table close parentheses equals stack C B with rightwards arrow on top

stack bold italic A bold italic B with bold rightwards arrow on top bold minus stack bold italic A bold italic C with bold rightwards arrow on top bold equals stack bold italic C bold italic B with bold rightwards arrow on top

c) Write negative 3 stack C B with rightwards arrow on top as a column vector.

Multiply all parts of the vector stack C B with rightwards arrow on top by negative 3

negative 3 stack C B with rightwards arrow on top equals negative 3 open parentheses table row cell negative 1 end cell row 8 end table close parentheses equals open parentheses table row cell negative 3 cross times negative 1 end cell row cell negative 3 cross times 8 end cell end table close parentheses

bold minus bold 3 stack bold C bold B with bold rightwards arrow on top bold equals stretchy left parenthesis table row 3 row cell negative 24 end cell end table stretchy right parenthesis

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    Jamie Wood

    Author: Jamie Wood

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.