Problem Solving using Vectors (Cambridge O Level Additional Maths)

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Problem Solving using Vectors

What problems may I be asked to solve involving vectors?

  • Showing that two lines or vectors are parallel
    • Two vectors are parallel if they are scalar multiples of each other
    • i.e.  bold a equals k bold b where k is a constant
      • See Vector Addition
  • Finding the midpoint of two (position) vectors
  • Showing that three points are collinear
    • Collinear describes points that lie on the same straight line
      • e.g.  The points open parentheses negative 2 comma space minus 2 close parentheses comma space open parentheses 3 comma space 3 close parentheses and open parentheses 8 comma space 8 close parentheses all lie on the line with equation y equals x
      • Vectors can be used to show this, and similar, results
  • Results concerned with geometric shapes
    • Shapes with parallel lines are often involved
      • e.g.  parallelogram, rhombus
    • These often include lines or vectors being split into ratios
      • e.g.  The point Q lies on the line P R such that stack P Q with rightwards arrow on top colon stack Q R with rightwards arrow on top equals 3 colon 1

How do I find the midpoint of two vectors?

  • If the point A has position vector bold a and the point B has position vector bold b
    • the position vector of the midpoint of A B is 1 half open parentheses bold a plus bold b close parentheses
  • This can be derived by considering
    • stack A B with rightwards arrow on top equals bold b minus bold a
      • using the result from Vector Addition 
    • If M is the midpoint of A B then
      • stack A M with rightwards arrow on top equals 1 half stack A B with rightwards arrow on top
    • Therefore, the position vector of the midpoint, stack O M with rightwards arrow on top is  
      • stack O M with rightwards arrow on top equals stack O A with rightwards arrow on top plus stack A M with rightwards arrow on top equals bold a plus 1 half open parentheses bold b minus bold a close parentheses
        stack O M with rightwards arrow on top equals 1 half open parentheses bold a plus bold b close parentheses 

How do I show three points are collinear?

  • Three points are collinear if they all lie on the same straight line
  • There are two ways to show this for three points, A comma space B and C say
    • Method 1
      Show that stack A B with rightwards arrow on top equals k stack A C with rightwards arrow on top where k is a constant
      i.e.  show that stack A B with rightwards arrow on top and stack A C with rightwards arrow on top are scalar multiples of each other
      • As the vectors are scalar multiples they will have the same direction (and so be parallel)
      • So as both vectors start at point A, they must be collinear
    • Method 2
      Show that stack A B with rightwards arrow on top equals k stack B C with rightwards arrow on top  AND  that point B lies on both the vectors stack A B with rightwards arrow on top and stack B C with rightwards arrow on top
  • Which method you should use will depend on the information given and how you happen to see the question

How do I solve problems involving geometric shapes?

  • Problems involving geometric shapes involve finding paths around the shape using known vectors
    • there will be many other vectors in the shape that are equal and/or parallel to the known vectors
  • The following grid is made up entirely of parallelograms, with the vectors bold a and bold b defined as marked in the diagram:

Vector parallelogram grid, Maths revision notes

  • Note the difference between "specific" and "general" vectors
    • The vector stack A B with rightwards arrow on top in the diagram is specific and refers only to the vector starting at A and ending at B
      • However, the vector bold a is a general vector
        • any vector the same length as stack A B with rightwards arrow on top and parallel to it is equal to bold a
        • e.g.  stack R S with rightwards arrow on top equals bold a
      • Vector bold b is also a general vector
        • e.g.  stack G L with rightwards arrow on top equals bold b 
    • There will also be vectors in the diagram that are the same magnitude but have the opposite direction to bold a or bold b
      • e.g.  stack O N with rightwards arrow on top equals negative bold a italic comma italic space italic space stack J E with italic rightwards arrow on top italic equals italic minus bold b
  • There are also many instances of the vector addition result stack F B with rightwards arrow on top equals bold b minus bold a
    • e.g.  stack P L with rightwards arrow on top equals bold b minus bold a
  • There are many scalar multiples of the vectors bold a or bold b
    • e.g.  stack F I with rightwards arrow on top equals 3 bold a italic comma italic space italic space stack I S with italic rightwards arrow on top equals 2 bold b italic comma italic space italic space stack Q E with italic rightwards arrow on top equals 3 begin italic style stretchy left parenthesis bold b minus bold a stretchy right parenthesis end style
  • Using a combination of these it is possible to describe a vector between any two points in terms of bold a and bold b

Examiner Tip

  • Diagrams are helpful in vector questions
    • If a diagram has been given, label it and add to it as you progress through a question
    • If a diagram has not been given, draw one, it does not need to be accurate!

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, 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 follow vector bold 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 plus stack B C with rightwards arrow on top plus space stack C D with rightwards arrow on top plus stack D E with rightwards arrow on top end cell row blank equals cell space bold italic a plus bold a plus bold a plus bold a end cell end table
 
stack A E with rightwards arrow on top space equals space 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 (bold b twice), and then from to (bold a three times).
 
table row cell stack G T with rightwards arrow on top end cell equals cell stack G L with rightwards arrow on top plus stack L Q with rightwards arrow on top plus stack Q R with rightwards arrow on top plus stack R S with rightwards arrow on top plus stack S T with rightwards arrow on top end cell row blank equals cell bold b plus bold b plus bold a plus bold a plus space bold a end cell end table
 
stack G T with rightwards arrow on top space equals space 3 bold a space plus space 2 bold b
 
c)
Point Z is such that it is midpoint of H M.
Find the vector stack P Z with rightwards arrow on top.
  
There are many ways to get from P to Z.
One option is to go from to (bold a twice), and then from R to Z (negative bold b one-and-a-half times).
 
table row cell stack P Z with rightwards arrow on top end cell equals cell stack P R with rightwards arrow on top plus stack R Z with rightwards arrow on top space space end cell row blank equals cell bold a plus bold a minus bold b minus 1 half bold b end cell end table
 
stack P Z with rightwards arrow on top equals 2 bold a minus 3 over 2 bold b

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.