GCSEMaths: FoundationEdexcelRevision NotesGeometry & MeasuresVectorsIntroduction to Vectors

Introduction to Vectors (Edexcel GCSE Maths: Foundation)

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Dan

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Basic Vectors

What are column vectors?

  • A column vector can be used to describe how to get from one point to another point
    • This is also called a translation vector
    • open parentheses table row 6 row 3 end table close parentheses means 6 units to the right and 3 units up

Column vector

How do I add and subtract column vectors?

  • Adding and subtracting vectors is done by looking at the top numbers and bottom numbers separately
  • To add column vectors
    • Add the top numbers together
    • Add the bottom numbers together
      • open parentheses table row 5 row 2 end table close parentheses plus open parentheses table row 3 row cell negative 1 end cell end table close parentheses equals open parentheses table row cell 5 plus 3 end cell row cell 2 plus open parentheses negative 1 close parentheses end cell end table close parentheses equals open parentheses table row 8 row 1 end table close parentheses
  • To subtract column vectors
    • Subtract the second top number from the first
    • Subtract the second bottom number from the first
      • open parentheses table row 5 row 2 end table close parentheses minus open parentheses table row 3 row cell negative 1 end cell end table close parentheses equals open parentheses table row cell 5 minus 3 end cell row cell 2 minus open parentheses negative 1 close parentheses end cell end table close parentheses equals open parentheses table row 2 row 3 end table close parentheses

How do I multiply a vector by a scalar?

  • A scalar is number not a vector
    • It does not have a direction
  • To multiply a column vector by a scalar
    • Multiply the top number by the scalar
    • Multiply the bottom number by the scalar
      • 3 open parentheses table row 2 row cell negative 1 end cell end table close parentheses equals open parentheses table row cell 3 cross times 2 end cell row cell 3 cross times open parentheses negative 1 close parentheses end cell end table close parentheses equals open parentheses table row 6 row cell negative 3 end cell end table close parentheses

How do I write an expression as a single column vector?

  • You need to follow the order of operations
    • 2 open parentheses table row 5 row 2 end table close parentheses plus 5 open parentheses table row 3 row cell negative 1 end cell end table close parentheses
  • STEP 1
    Multiply each vector by the scalar in front of it
    • open parentheses table row cell 2 cross times 5 end cell row cell 2 cross times 2 end cell end table close parentheses plus open parentheses table row cell 5 cross times 3 end cell row cell 5 cross times open parentheses negative 1 close parentheses end cell end table close parentheses equals open parentheses table row 10 row 4 end table close parentheses plus open parentheses table row 15 row cell negative 5 end cell end table close parentheses
  • STEP 2
    Add or subtract the new column vectors
    • open parentheses table row cell 10 plus 15 end cell row cell 4 plus open parentheses negative 5 close parentheses end cell end table close parentheses equals open parentheses table row 25 row cell negative 1 end cell end table close parentheses

Worked example

bold a equals open parentheses table row p row 3 end table close parentheses and bold b equals open parentheses table row cell negative 2 end cell row 1 end table close parentheses.

Given that 2 bold a plus 3 bold b equals open parentheses table row 4 row q end table close parentheses, find the value of p and the value of q.

Write the left-side side as one vector
Multiple each vector by the scalar in front of it

open parentheses table row cell 2 p end cell row 6 end table close parentheses plus open parentheses table row cell negative 6 end cell row 3 end table close parentheses equals open parentheses table row 4 row q end table close parentheses

Add the vectors together

open parentheses table row cell 2 p minus 6 end cell row 9 end table close parentheses equals open parentheses table row 4 row q end table close parentheses

The top components are equal
Form and solve an equation

table row cell 2 p minus 6 end cell equals 4 row cell 2 p end cell equals 10 row p equals 5 end table

The bottom components are equal

9 equals q

bold italic p bold equals bold 5 and bold italic q bold equals bold 9

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.Question points on grid, IGCSE & GCSE Maths revision notes

(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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    Dan

    Author: Dan

    Expertise: Maths

    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.