Working with Vectors (OCR GCSE Maths)

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Naomi C

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Naomi C

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Finding Vector Paths

Finding paths in vector diagrams

  • It is important to be able to describe vectors by following paths through a geometric diagram
  • The following grid is made up entirely of parallelograms, with the vectors  and  defined as marked in the diagram:

Vector parallelogram grid, IGCSE & GCSE 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 a is a general vector - any vector the same length as stack A B with rightwards arrow on top and pointing in the same direction is equal to a
    • Similarly, any vector the same length as stack A F with rightwards arrow on top and pointing in the same direction is equal to b
  • Following a vector in the "wrong" direction (i.e. from head to tail instead of from tail to head) makes a general vector negative
    • So in the diagram above stack A B with rightwards arrow on top equals a, but stack B A with rightwards arrow on top equals negative a
    • Similarly stack A F with rightwards arrow on top equals b and stack F A with rightwards arrow on top equals negative b
  • Note in particular the vector stack F B with rightwards arrow on top:Vector paths vector FB 01, IGCSE & GCSE Maths revision notes
  • Getting from point to point  we have to go the 'wrong way' down  and then the 'right way' along

Vector paths vector FB 02, IGCSE & GCSE Maths revision notes

  • It follows that:
    • stack F B with rightwards arrow on top equals negative b plus a equals a minus b
  • and of course then
    • stack B F with rightwards arrow on top equals negative stack F B with rightwards arrow on top equals negative open parentheses a minus b close parentheses equals b minus a

  • Keeping those things in mind, it is possible to describe any vector that goes from one point to another in the above diagram in terms of a and b

Examiner Tip

  • Adding and subtracting vectors follows all the same rules as adding and subtracting letters like x and y in algebra (this includes collecting like terms).
  • It doesn't matter exactly what path you follow through a diagram from starting point to ending point – as long as you add and subtract the general vectors correctly along the path you use, you will get the correct answer.

Worked example

The following diagram consists of a grid of identical parallelograms.

Vectors a and b are defined by bold italic a space equals space stack A B with rightwards arrow on top and bold italic 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 a and 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 italic a space plus bold space bold italic a space space plus space bold italic a space plus space bold italic 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 italic 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 italic b space plus space bold italic b space plus space bold italic a space plus space bold italic a space plus space bold italic 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 italic a bold space bold plus bold space bold 2 bold italic 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 italic b space plus space bold italic b space minus space bold italic a space minus space bold italic a space minus space bold italic a bold space bold minus bold space bold italic 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 italic b bold space bold minus bold space bold 4 bold italic a
-4+ 2also acceptable

 

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Vector Proof

What are vector proofs?

  • In vector proofs we use vectors, along with a few key ideas, to prove that things are true in geometrical diagrams
  • Problem solving with vectors involves using these vector proofs to help us to find out additional information

Parallel vectors

  • Two vectors are parallel if and only if one is a multiple of the other
  • This tends to appear in vector proofs in the following ways:
    • If you find in your workings that one vector is a multiple of the other, then you know that the two vectors are parallel – you can then use that fact in the rest of the proof
    • If you need to show that two vectors are parallel, then all you need to do is show that one of the vectors multiplied by some number is equal to the other one
    • E.g. If stack A B with rightwards arrow on top equals a plus 2 b and stack C D with rightwards arrow on top equals 3 a plus 6 b, then stack C D with rightwards arrow on top equals 3 open parentheses a plus 2 b close parentheses equals 3 stack A B with rightwards arrow on top, therefore stack A B with rightwards arrow on top and stack C D with rightwards arrow on top are parallel

Points on a straight line

  • Often you are asked to show in a vector proof that three points lie on a straight line (ie that they are collinear)
  • To show that three points AB and C lie on a straight line,
    • show that the vectors connecting the three points are parallel,
      • for example, show that stack A B with rightwards arrow on top is a multiple of (and therefore parallel to) stack A C with rightwards arrow on top, or that stack A B with rightwards arrow on top is a multiple of (and therefore parallel) to stack B C with rightwards arrow on top
    • as those two vectors are parallel and they share a common point it means that the three lines form a straight line

Collinear and Non-collinear Points, IGCSE & GCSE Maths revision notes

Vectors divided in ratios

  • Be careful turning ratios into fractions in vector proofs!
  • If a point  divides a line segment  in the ratio p : q, then: 

stack A X with rightwards arrow on top equals fraction numerator p over denominator p plus q end fraction stack A B with rightwards arrow on top and stack X B with rightwards arrow on top equals fraction numerator q over denominator p plus q end fraction stack A B with rightwards arrow on top

    • eg. In the following diagram, the point  divides  in the ratio 3: 5:

Vector line divided in ratio, IGCSE & GCSE Maths revision notes

Therefore

stack A X with rightwards arrow on top equals 3 over 8 stack A B with rightwards arrow on top and stack X B with rightwards arrow on top equals 5 over 8 stack A B with rightwards arrow on top

Worked example

The diagram shows trapezium OABC.

stack O A with rightwards arrow on top space equals space 2 bold italic a

stack O B with rightwards arrow on top space equals space bold italic c

AB is parallel to OC, with stack A B with rightwards arrow on top space equals space 3 stack O C with rightwards arrow on top.

Question Vector Trapezium, IGCSE & GCSE Maths revision notes

 

a)
Find expressions for vectors stack O B with rightwards arrow on top and stack A C with rightwards arrow on top in terms of a and c
 
stack A B with rightwards arrow on top space equals space 3 stack O C with rightwards arrow on top and stack O C with rightwards arrow on top space equals bold space bold italic c  so  stack A B with rightwards arrow on top space equals space 3 bold italic c.
 

table attributes columnalign right center left columnspacing 0px end attributes row cell stack O B with rightwards arrow on top space end cell equals cell space stack O A with rightwards arrow on top space plus space stack A B with rightwards arrow on top space end cell row blank equals cell space 2 bold italic a space plus space 3 bold italic c end cell end table

stack bold italic O bold italic B with bold rightwards arrow on top bold space bold equals bold space bold 2 bold italic a bold space bold plus bold space bold 3 bold italic c

table attributes columnalign right center left columnspacing 0px end attributes row cell stack A C with rightwards arrow on top space end cell equals cell space stack A O with rightwards arrow on top space plus space stack O C with rightwards arrow on top end cell row blank equals cell space minus stack O A with rightwards arrow on top space plus space stack O C with rightwards arrow on top end cell row blank equals cell space minus 2 bold italic a space plus space bold italic c end cell end table

stack bold italic A bold italic C with bold rightwards arrow on top bold space bold equals bold space bold italic c bold space bold minus bold space bold 2 bold italic a

 

b)
Point P lies on AC such that AP : PC = 3 : 1.
Find expressions for vectors stack A P with rightwards arrow on top and stack O P with rightwards arrow on top in terms of a and c
 
AP : PC = 3 : 1 means that  stack A P with rightwards arrow on top space equals space fraction numerator 3 over denominator 3 plus 1 end fraction stack A C space with rightwards arrow on top space equals space 3 over 4 stack A C space with rightwards arrow on top space.
 

table attributes columnalign right center left columnspacing 0px end attributes row cell stack A P with rightwards arrow on top space end cell equals cell space 3 over 4 space stack A C with rightwards arrow on top space equals space 3 over 4 open parentheses negative 2 bold italic a space plus space bold italic c close parentheses end cell end table

stack bold italic A bold italic P with bold rightwards arrow on top bold space bold equals bold space bold minus bold 3 over bold 2 bold italic a bold space bold plus bold space bold 3 over bold 4 bold italic c

table attributes columnalign right center left columnspacing 0px end attributes row cell stack O P with rightwards arrow on top space end cell equals cell space stack O A with rightwards arrow on top space plus space stack A P with rightwards arrow on top end cell row blank equals cell space 2 bold italic a space plus space begin bold style stretchy left parenthesis straight minus straight 3 over straight 2 a space plus space straight 3 over straight 4 c stretchy right parenthesis end style end cell end table

stack bold italic O bold italic P with bold rightwards arrow on top bold space bold equals bold space bold 1 over bold 2 bold italic a bold space bold plus bold space bold 3 over bold 4 bold italic c

 

c)
Hence, prove that point P lies on line OB, and determine the ratio stack O P with rightwards arrow on top space colon space stack P B with rightwards arrow on top.
 
To show that O, P, and B are colinear (lie on the same line), note that  stack O P with rightwards arrow on top space equals space 1 half bold italic a space plus space 3 over 4 bold italic c bold space equals space 1 fourth open parentheses 2 bold italic a space plus space 3 bold italic c close parentheses.
 

table row cell stack O P with rightwards arrow on top space end cell equals cell space 1 fourth open parentheses 2 bold italic a space plus space 3 bold italic c close parentheses space equals space 1 fourth stack O B with rightwards arrow on top end cell end table

stack bold italic O bold italic P with bold rightwards arrow on top bold space bold equals bold space bold 1 over bold 4 stack bold O bold B with bold rightwards arrow on top therefore OP is parallel to OB and so P must lie on the line OB 

If table attributes columnalign right center left columnspacing 0px end attributes row cell stack O P with rightwards arrow on top space end cell equals cell space 1 fourth stack O B with rightwards arrow on top space end cell end table then table attributes columnalign right center left columnspacing 0px end attributes row cell stack P B with rightwards arrow on top space end cell equals cell space 3 over 4 stack O B with rightwards arrow on top space end cell end table

stack bold italic O bold italic P with bold rightwards arrow on top space bold colon space stack bold italic P bold italic B with bold rightwards arrow on top space bold equals bold space bold 1 bold space bold colon bold space bold 3

   

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Naomi C

Author: Naomi C

Expertise: Maths

Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.