IGCSEMathsCambridge (CIE)International MathsExtendedRevision NotesVectors & TransformationsVectorsRepresenting Vectors as Diagrams

Representing Vectors as Diagrams (Cambridge (CIE) IGCSE International Maths)

Revision Note

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

Vector Diagrams

How can I represent a vector visually?

  • A vector has both a size (magnitude) and a direction

    • You need to draw a line to show the size of the vector

    • You also need to draw an arrow to show the direction of the vector

Magnitude and direction of a vector

 

  • Vectors are written in bold when typed to show that they are a vector and not a scalar

    • When writing a vector in an exam you should underline the letter to show it is a vector

    • bold italic a when typed and bottom enclose a when handwritten

      • You will not lose marks if you forget to underline vectors

  • If a vector starts at A and ends at B we can write it as stack A B with rightwards arrow on top

    • Here the arrow will point toward B

    • Vector stack B A with rightwards arrow on top will have the same length but point toward A

Vector between two points

How do I draw a vector on a grid?

  • You can draw a vector anywhere on a grid

    • Just make sure it has the correct length and the correct direction

  • To draw the vector bold a equals open parentheses table row 3 row 4 end table close parentheses

    • Pick a point on the grid and draw a dot there

    • Count 3 units to the right and 4 units up and draw another dot

    • Draw a line between the two dots

    • Put an arrow on the line pointing toward the second dot

  • Look out for negatives and zeroes

    • bold b equals open parentheses table row 2 row cell negative 4 end cell end table close parentheses  goes 2 to the right and 4 down

    • bold c equals open parentheses table row 2 row 0 end table close parentheses goes 2 to the right but does not go up or down

Vectors on a grid

What happens when I multiply a vector by a scalar?

  • When you multiply a vector by a positive scalar:

    • The direction stays the same

    • The length of the vector is multiplied by the scalar

  • For example, bold a equals open parentheses table row 4 row cell negative 2 end cell end table close parentheses

    • 2 bold a equals open parentheses table row 8 row cell negative 4 end cell end table close parentheses will have the same direction but double the length

    • 1 half bold a equals open parentheses table row 2 row cell negative 1 end cell end table close parentheses will have the same direction but half the length

Multiplying vectors by a scalar

  • When you multiply a vector by a negative scalar:

    • The direction is reversed

    • The length of the vector is multiplied by the number after the negative sign

  • For example, bold a equals open parentheses table row 4 row cell negative 2 end cell end table close parentheses

    • negative bold a equals open parentheses table row cell negative 4 end cell row 2 end table close parentheses will be in the opposite direction and its length will be the same

    • negative 2 bold a equals open parentheses table row cell negative 8 end cell row 4 end table close parentheses will be in the opposite direction and its length will be doubled

Multiplying a vector by a negative scalar

What happens when I add or subtract vectors?

  • To draw the vector bold a plus bold b

    • Draw the vector bold a

    • Draw the vector bold b starting at the endpoint of bold a

    • Draw a line that starts at the start of bold a and ends at the end of bold b

  • To draw the vector bold a minus bold b 

    • Draw the vector bold a

    • Draw the vector negative bold b  starting at the endpoint of bold a

    • Draw a line that starts at the start of bold a  and ends at the end of negative bold b

Adding and subtracting vectors

Worked Example

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

Question points with vectors, IGCSE & GCSE Maths revision notes

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

stack bold A bold B with bold rightwards arrow on top bold equals stretchy left parenthesis table row 6 row 2 end table stretchy right parenthesis  

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

stack bold A bold C with bold rightwards arrow on top bold equals stretchy left parenthesis table row 7 row cell negative 6 end cell end table stretchy right parenthesis  

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

stack bold C bold B with bold rightwards arrow on top bold equals stretchy left parenthesis table row cell negative 1 end cell row 8 end table stretchy right parenthesis

   

(b)

Without using any calculations, explain why stack A B with rightwards arrow on top plus stack B C with rightwards arrow on top plus stack C A with rightwards arrow on top equals open parentheses table row 0 row 0 end table close parentheses.

The vector goes from A to B, then from B to C, then from C back to A

The vector returns to its starting point

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    Jamie Wood

    Author: Jamie Wood

    Expertise: Maths

    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.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.