Vector Proof (Edexcel IGCSE Maths A (Modular))

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Vector Proof

What are vector proofs?

  • Vectors can be used to prove things that are true in geometrical diagrams

    • Vector proofs can be used to find additional information that can help us to solve problems

How do I know if two vectors are parallel?

  • Two vectors are parallel if one is a scalar multiple of the other

    • This means if b is parallel to a, then b = ka

      •  where k is a constant number (scalar)

  • For example, bold a equals open parentheses table row 1 row 3 end table close parentheses and bold b equals open parentheses table row 2 row 6 end table close parentheses 

    • open parentheses table row 2 row 6 end table close parentheses equals 2 cross times open parentheses table row 1 row 3 end table close parentheses so bold b equals 2 bold a

    • b is a scalar multiple of a, so b is parallel to a

  • If the scalar multiple is negative, then the vectors are parallel and in opposite directions

    • bold c equals open parentheses table row cell negative 3 end cell row cell negative 9 end cell end table close parentheses equals negative 3 bold a

      • c is parallel to a and in the opposite direction

  • If two vectors factorise with a common bracket, then they are parallel

    • They can be written as scalar multiples 

  • For example

    • 9a + 6factorises to 3(3a + 2b)

    • 12a + 8factorises to 4(3a + 2b)

    • This means 12 bold a plus 8 bold b equals 4 over 3 open parentheses 9 bold a plus 6 bold b close parentheses

      • so they are scalar multiples of each other

      • and therefore parallel

How do I know if three points lie on a straight line?

  • You may be asked to prove that three points lie on a straight line

    • Points that lie on a straight line are collinear

  • To show that the points A , and are collinear

    • prove that two line segments are parallel

    • and show that there is at least one point that lies on both segments

      • This makes them parallel and connected (not parallel and side-by-side)

  • For example, if you show that stack B C with rightwards arrow on top equals 2 stack A B with rightwards arrow on top then

    • the line segments AB  and BC  are parallel

    • and they have a common point,

      • So A , and must be collinear

  • Similarly, stack A C with rightwards arrow on top equals 3 stack A B with rightwards arrow on top means AC  and AB  are parallel

    • and they have a common point, A

      • so A , and must be collinear

If A, B, C are collinear, AB is parallel to AC and BC

How do I use ratios in vector paths?

Vector line divided into a ratio
  • Convert ratios into fractions

  • In the example shown, if A X space colon space X B equals 3 colon 5 then

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

    • stack X B with italic rightwards arrow on top equals 5 over 8 stack A B with rightwards arrow on top

      • The ratio 3:5 has 3 + 5 = 8 parts

  • Always check which ratio you are being asked for

    • stack A X with italic rightwards arrow on top equals 3 over 5 stack X B with rightwards arrow on top

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

Worked Example

The diagram shows trapezium OABC.

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

stack O B with rightwards arrow on top equals bold c

AB is parallel to OC, with stack A B with rightwards arrow on top equals 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 equals 3 stack O C with rightwards arrow on top and stack O C with rightwards arrow on top equals bold c  so  stack A B with rightwards arrow on top equals 3 bold italic c.

table row cell stack O B with rightwards arrow on top end cell equals cell stack O A with rightwards arrow on top plus stack A B with rightwards arrow on top space end cell row blank equals cell 2 bold a plus 3 bold c end cell end table

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

table row cell stack A C with rightwards arrow on top end cell equals cell stack A O with rightwards arrow on top plus stack O C with rightwards arrow on top end cell row blank equals cell negative stack O A with rightwards arrow on top plus stack O C with rightwards arrow on top end cell row blank equals cell negative 2 bold a plus bold c end cell end table

stack bold italic A bold italic C with bold rightwards arrow on top bold equals bold italic c bold minus 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 equals fraction numerator 3 over denominator 3 plus 1 end fraction stack A C with rightwards arrow on top equals 3 over 4 stack A C with rightwards arrow on top space

table row cell stack A P with rightwards arrow on top end cell equals cell 3 over 4 space stack A C with rightwards arrow on top equals 3 over 4 open parentheses negative 2 bold a space plus space bold c close parentheses end cell end table

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

table row cell stack O P with rightwards arrow on top end cell equals cell stack O A with rightwards arrow on top plus stack A P with rightwards arrow on top end cell row blank equals cell 2 bold a plus stretchy left parenthesis negative 3 over 2 bold a plus 3 over 4 bold c stretchy right parenthesis end cell end table

stack bold italic O bold italic P with bold rightwards arrow on top bold equals bold 1 over bold 2 bold a bold plus bold 3 over bold 4 bold 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 equals 1 half bold a plus 3 over 4 bold c equals 1 fourth open parentheses 2 bold a plus 3 bold c close parentheses

table row cell stack O P with rightwards arrow on top end cell equals cell 1 fourth open parentheses 2 bold a plus 3 bold c close parentheses equals 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 O B with italic rightwards arrow on top therefore OP is parallel to OB
and so P must lie on the line OB

If table row cell stack O P with rightwards arrow on top end cell equals cell 1 fourth stack O B with rightwards arrow on top space end cell end table then table row cell stack P B with rightwards arrow on top end cell equals cell 3 over 4 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 space bold colon bold space stack bold italic P bold italic B with bold rightwards arrow on top bold equals bold 1 bold space bold colon bold space bold 3

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Jamie Wood

Author: Jamie Wood

Expertise: Maths

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

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.

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