Problem Solving using Vectors (Cambridge (CIE) IGCSE Additional Maths) : Revision Note

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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 Tips and Tricks

  • 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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