Working with Vectors (Edexcel GCSE Maths: Foundation)

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Finding Vector Paths

How do I find the vector between two points?

  • You need to be able to find the vector using a grid
    • The grid will be made up of parallelograms or equilateral triangles
  • The following grid is made up entirely of parallelograms, with the vectors a and b defined as marked in the diagram:
    • Any vector that goes horizontally to the right along a side of a parallelogram will be equal to a
    • Any vector that goes up diagonally to the right along a side of a parallelogram will be equal to b

Vectors on a grid of parallelograms

  • To find the vector between two points
    • Count how many times you need to go horizontally to the right
      • This will tell you how many a's are in your answer
    • Count how many times you need to go up diagonally to the right
      • This will tell you how many b's are in your answer
    • Add the a's and b's together
      • e.g. stack A R with rightwards arrow on top equals 2 bold a plus 3 bold b
  • You will have to put a negative in front of the vector if it goes in the opposite direction
    • -a is one length horizontally to the left
    • -b is one length down diagonally to the left
      • e.g. stack F B with rightwards arrow on top equals negative bold b plus bold a

Vector paths on a grid

Examiner Tip

  • Adding and subtracting vectors follows all the same rules as adding and subtracting letters like a and b in algebra (this includes collecting like terms).
  • Always look for the easiest path between two points
    • Go as far as you can in one direction
    • And then use the other direction

Worked example

The following diagram consists of a grid of identical parallelograms.

Vectors bold a and bold b are defined by bold a space equals space stack A B with rightwards arrow on top and bold b bold space equals space stack A F with rightwards arrow on top.

 

Vector parallelogram grid, IGCSE & GCSE Maths revision notes

Write the following vectors in terms of bold a and bold b.

a)
stack A E with rightwards arrow on top
  
To get from A to E we need to follow vector a four times to the right.
 
table row cell stack A E with rightwards arrow on top space end cell equals cell space stack A B with rightwards arrow on top space plus thin space stack B C with rightwards arrow on top space plus space stack C D with rightwards arrow on top space plus space stack D E with rightwards arrow on top end cell row blank equals cell space bold a space plus bold space bold a space space plus space bold a space plus space bold a end cell end table
 
stack bold italic A bold italic E with bold rightwards arrow on top bold space bold equals bold space bold 4 bold a 

b)
stack G T with rightwards arrow on top
  
There are many ways to get from G to T. One option is to go from to (twice), and then from to (a three times).
 
table row cell stack G T with rightwards arrow on top space end cell equals cell space stack G L with rightwards arrow on top space plus thin space stack L Q with rightwards arrow on top space plus space stack Q R with rightwards arrow on top space plus space stack R S with rightwards arrow on top space plus space stack S T with rightwards arrow on top end cell row blank equals cell bold b space plus space bold b space plus space bold a space plus space bold a space plus space bold a end cell end table
 
stack bold italic G bold italic T with bold rightwards arrow on top bold space bold equals bold space bold 3 bold a bold space bold plus bold space bold 2 bold b
 
c)
stack E K with rightwards arrow on top
  
There are many ways to get from E to K. One option is to go from E to O (twice), and then from O to K ( -a four times).
 
table row cell stack E K with rightwards arrow on top space end cell equals cell space stack E J with rightwards arrow on top space plus thin space stack J O with rightwards arrow on top space plus space stack O N with rightwards arrow on top space plus space stack N M with rightwards arrow on top space plus space stack M L with rightwards arrow on top space plus space stack L K with rightwards arrow on top space space end cell row blank equals cell bold b space plus space bold b space minus space bold a space minus space bold a space minus space bold a bold space bold minus bold space bold a end cell end table
 
stack bold italic E bold italic K with bold rightwards arrow on top bold space bold equals bold space bold space bold 2 bold b bold space bold minus bold space bold 4 bold a
negative 4 bold a space plus space 2 bold b also acceptable

 

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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.