GCSE Mathematics for Edexcel Student Book Sample by CUPUKschools

Blended digital and print resources specifically created for the new Edexcel GCSE Mathematics specification, available from early 2015.

Brighter Thinking

Brighter thinking for the new curriculum: • Written by an experienced author team of teachers, partners and advisers. • Rich digital content to engage and motivate learners. • Differentiated resources to support all abilities. • Progression and development at the heart of all our resources.

For more information or to speak to your local sales consultant, please contact us: www.cambridge.org/ukschools ukschools@cambridge.org 01223 325 588 CUPUKschools

MATHEMATICS GCSE for Edexcel Sample

Written from draft specification

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Foreword I’m delighted to introduce this innovative GCSE suite of resources from Cambridge University Press, the respected publishing house with a rich history in the field of mathematics and mathematics educational publishing. At this time of many changes – to the curriculum, assessment and professional development – it’s important to choose wisely. The new suite of Cambridge University Press mathematics resources have been designed specifically to meet the needs of teachers and students, and will support students in the successful application of mathematics in GCSE examinations and beyond. The new GCSE specifications will have a strong focus on problem-solving. Problem-solving is at the heart of mathematics, and to be successful at GCSE students will need to be completely familiar with problem-solving strategies, as well as confident enough to be able to tackle non-routine problems. This set of resources specifically promotes progress in the three areas of problem-solving, reasoning and fluency, and features comprehensive tracking so that you will know where to focus your attention and students can monitor their own progress. The Teacher’s Resource offers ideas and advice on planning, and includes teaching suggestions for both Foundation and Higher tiers so that you can mix and match where appropriate. These are complemented by question and assessment banks that not only support knowledge in specific areas, but also offer tasks that require your students to make connections within and across areas of the curriculum – a vital skill for success. NRICH is, of course, committed to supporting teachers who want their students to think and behave like mathematicians, which is why we are happy to work with Cambridge University Press on this project. With built-in support for independent learning and contextual features to make mathematics relevant, I am sure these resources will prove invaluable as you implement the inspirational aims of the new curriculum. Lynne McClure, Series Consultant and Director at NRICH, Cambridge

Working in partnership with NRICH

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Author profiles Karen Morrison Karen is an experienced teacher and teacher-trainer who now works as an author and materials developer. She is the author of a number of mathematics courses. Karen has travelled extensively and has worked collaboratively with teachers and authors in the UK, Caribbean, Middle East, Asia and Africa.

Julia Smith Julia is the Director of Mathematics at Tendring Enterprise Studio School. She is also an Initial Teacher Trainer leading on mathematics with Suffolk/Norfolk ITT and the SSAT, and an outstanding classroom practitioner with management experience within secondary and FE. Awarded the Goldsmith’s Award in 2013, Julia has worked with the Open University as a Mathematics Portal Manager, is a specialist on the Functional Mathematics Review Panel and has authored a number of secondary mathematics titles.

Pauline McLean Pauline worked at Norfolk Local Authority as a Mathematics Advisor and has been a mathematics teacher for over 20 years. She worked for Capita’s school improvement service and is connected to the Learning Skills Trust. Pauline works on developing mathematics pedagogies and is Ofsted trained.

Rachael Horsman Rachael currently works across a group of schools for Comberton Village Academy Trust as an Advanced Skills Teacher and Specialist Leader of Education, supporting teaching staff and Heads of Department, delivering training and working closely with national organisations. Recently appointed as Lead for the Cambridge Maths Hub, Rachael also acts as Chair for the 11–16 sub-committee and Teaching Committee. She is a member of the Council for the Mathematical Association (MA) where she delivers CPD workshops and training sessions, helps to form responses to government consultations and develops resources for publication. She has also held positions as Director of Maths, Primary and Secondary Liaison and Assistant Head.

Nick Asker Nick is a Mathematics Advisor for Improve Maths. Nick started teaching mathematics in 1982 and has spent almost 20 years teaching in a variety of settings including secondary schools and all-age special schools. Nick has been a Head of Department and a Local Authority Mathematics Advisor, as well as a Regional Advisor for the National Strategies. He supports mathematics departments to develop their teaching through plan-teach-review cycles. He holds an NPQH and is qualified to lead inspections on behalf of Ofsted.

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Our offering for Edexcel At Cambridge University Press, we are driven by a simple goal: to create resources that teachers and students need to ignite a curiosity and love for learning. As England enters a new educational chapter, we are publishing a comprehensive suite of blended print and digital mathematics resources specifically written for the new Edexcel GCSE Mathematics specification, available from early 2015. Written by an experienced author team of teachers, partners and advisers, our mathematics resources have a strong focus on the development of problem-solving skills, fluency and mathematical reasoning. With progression at the heart, we provide the tools for students to become independent learners, encouraging them to understand the relevance of mathematics in the real world by demonstrating its application, purpose and context. With rich digital content to engage and motivate learners, and differentiated to support all abilities, our simple and affordable resources build on subject knowledge, and help prepare students for achievement in the new GCSE specification.

Foundation

Higher

Student Book A print Student Book bundled with GCSE Mathematics Online, covering all knowledge, understanding and skills for the Foundation and Higher courses. GCSE Mathematics Online An interactive learning, teaching and assessment tool for students and teachers that includes interactive lessons, tasks, questions, quizzes and widgets. Teacher’s Resource Everything necessary for teachers to plan and deliver the specification. Problem-solving Book Supplementary practice to help students develop their problemsolving skills. Exam Preparation A digital assessment and progress tracking tool providing practice and preparation for the linear exams. Homework Book Containing a variety of questions corresponding to the chapters in the Student Book, it includes problem-solving and mathematical reasoning questions. Our inclusive print Student Book and GCSE Mathematics Online bundle offers a sophisticated and cost‑effective solution, including everything necessary for the effective teaching and learning of the new GCSE specification in one package.

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Our resources Student Book

GCSE Mathematics Online

Bundled with our digital GCSE Mathematics Online resource, our print Student Books for the Higher and Foundation tiers have been specifically written for the new GCSE Mathematics curriculum.

Developed specifically for the new GCSE Mathematics curriculum, GCSE Mathematics Online for both the Foundation and Higher tiers is available as a standalone product or as part of our print Student Book and digital bundle. This enhanced digital learning, teaching and assessment resource offers a progressive course structure.

With a strong focus on the development of problemsolving skills, fluency and mathematical reasoning, our resources provide the tools to help students reach their potential, whilst developing their confidence and enjoyment of mathematics. With progress at the heart, our GCSE Mathematics Student Books feature: • examples to engage students and show the importance, relevance and application of mathematics in the real world • Launchpad, a feature at the beginning of each chapter directing students to their best starting point to optimise progress and promote independent learning • a variety of practice, investigatory, reasoning and problem-solving questions and exercises, that gradually increase in difficulty and link together to develop fluency of mathematics • tips on calculator use, vocabulary and revision • Work it out features to discover and correct any misconceptions • checklists, end-of-chapter review exercises, and cross-topic question sets to provide a range of assessment opportunities.

GCSE Mathematics Online includes comprehensive provision for both summative and formative assessment, with a range of differentiated preset quizzes and a test generator to compile your own assessments. Inbuilt reporting allows you to track students’ work and progression, so areas of strength and weakness can be easily identified. GCSE Mathematics Online features: • a flexible suite of resources for all types of classroom set-up, including material for interactive whiteboards and projectors • resources organised by topics and lessons with explanatory notes • investigations and interactive widgets to visually demonstrate concepts • walkthroughs that take students through a question step-by-step, with feedback • quick-fire quizzes with leader boards providing an opportunity for question practice • levelled questions that assess understanding of the topic • the ability to create and set tasks for lessons, auto-mark, report on tests and homework and review student performance.

Access free transition support materials from GCSE Mathematics Online now, visit www.gcsemaths.cambridge.org

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Teacher’s Resource

Homework Book

Our FREE digital Teacher’s Resource, for both Foundation and Higher Tier, contains practical support and guidance for delivering the new GCSE Mathematics specification.

Our Higher and Foundation GCSE Mathematics Homework Books are affordable, standalone resources and ideal companions to the Student Books.

Mapped to the Student Books, this extensive free resource is full of teaching ideas and advice. With notes on every Student Book chapter, it also links to the Problem-solving Books and Homework Books. Our Teacher’s Resource features: • chapter notes that cover prerequisite learning and offer ideas to introduce topics, guidance on activities in the Student Books, ways to address common errors, and connections to past and future learning

Each homework exercise corresponds to a section of the relevant Student Book and offers a further set of questions for practice and consolidation. They contain a breadth and depth of questions covering a variety of skills, including problem-solving and mathematical reasoning, as well as extensive drill questions. Answers to all questions are available for free. Our Homework Books: • offer practice questions for students across the full GCSE Mathematics curriculum

• suggested thought-provoking questions, starters, plenaries and enrichment tasks

• can be used in the classroom or set as homework

• photocopiable sheets to save time

• contain additional questions not found in the Student Books

• information about changes to GCSE and a topic map to help plan your route through the course

• are ideally sized for school bags.

• ideas for revision activities and advice on home learning • worked solutions to the mixed questions in the Student Books • a literacy chapter with exercises to practise extracting information from text and working out what mathematics needs to be done.

GCSE Mathematics Assessment and Progress Tracker Our new Test and Assessment tool is coming soon. Mapped to the Student Books, it is suitable for classwork, homework and revision, as well as both summative and formative assessment.

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Problem-solving Book Our Higher and Foundation GCSE Mathematics Problem-solving Books contain a variety of questions for students to develop their problem-solving and reasoning skills within the context of the new GCSE curriculum. Suitable for students at all levels, these resources will stretch the more able and support those who need it. Questions with worked solutions will help students develop the reasoning, interpreting, estimating and communication skills required to help them effectively solve problems. Our Problem-solving Books: • teach the skills necessary for the new AO2 and AO3 that now have a stronger emphasis in exams • introduce themes to guide students through the chapters, prompting different ways of thinking and allowing students to consider alternative techniques • offer a variety of contexts to develop problem‑solving and reasoning techniques • include questions with star ratings to indicate difficulty levels, so they can be used by a wide range of students • covers multiple topics and includes colour-coded questions to prepare students for the synoptic nature of the exams • provide fully-worked solutions and explanations.

Want to explore further? As well as Student Book sample chapters included here, you can browse samples of the additional GCSE Mathematics components, such as the Problem-solving Book and Teacher’s Resource, online at

www.cambridge.org/ukschools

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Using the Student Book

What are you learning? Short objectives to show the content and mathematical skills covered in the chapter.

Tip General hints and reminders to aid students, sometimes including questions to prompt students’ knowledge.

Using mathematics Shows how the content of the chapter relates to the world of work, making maths relevant beyond the classroom.

Before you start… Lists the prerequisite knowledge for the chapter. Includes questions on each concept to assess whether students can recall the necessary knowledge and skills.

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Launchpad A flow chart of concepts covered in the chapter. Each step tests a different concept level, progressing from least to most difficult. Students can assess their most appropriate starting point in the chapter, encouraging them to take ownership of their learning.

4 Functions and sequences

Assess your starting point using the Launchpad STep 1 1

a What are the next three numbers in the sequence 23, 35, 47, …? b What is the rule for finding the next term in this sequence?

✓

Go to Section 1: Sequences and patterns

STep 2 2

a What are the 10th, 20th and 100th terms in the sequence 3n 2 1? b What is the expression for the nth term of a sequence that starts 21, 2, 5, 8, …?

✓

Go to Section 2: Finding the nth term

STep 3 3 4

Draw an input-output flow diagram for the instruction ‘multiply the input number by 2 and then subtract 4’. The rule for generating a sequence is y 5 x 1 3. List the first five terms in the sequence.

✓

Go to Section 3: Functions

STep 4 5

What is special about the sequence 2, 4, 7, 11, … ?

✓

Go to Section 4: Special sequences

Go to Chapter review

151

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14 Functions and sequences

21, 4, 9, 14, … 21, 6, 13, 20, …

periment.

three-week period.

Using the Student Book

ek?

to be true.

Week

Height

1 2 3

4.5 cm 6.7 cm 8.9 cm

Worked example

4 Functions and sequences

WorKed exAmple 1

Teaching through set questions with exemplar solutions, complete with commentaries.

e to seat different

Table

Seats

‘I use term-to-term rules in my job. I know that each row of bricks will have three fewer bricks than the row below it, so I can work out how many bricks I need in each row.’

Key vocabulary

a line?

Key terms are highlighted for ease of learning, and gathered in a glossary at3 the back of each Student Book for 8 quick reference.

uence?

(Bricklayer)

d 7n 2 3

The first row of the wall has 57 bricks. Use a term-to-term rule to find the number of bricks in the next three rows. The number of bricks decreases by 3 for each new row.

The term-to-term rule is ‘subtract three’.

57 in the first row

ther.

add 1’ are examples of

Key vocabulary function:

ultiply by 2’ can be

a set of instructions for changing one number (the input) into another number (the output).

57 − 3 5 54 in the second row

Subtract 3 from 57 to get the next term.

54 − 3 5 51 in the third row 51 − 3 5 48 in the fourth row

Continue to do this for the next two terms. 4 Functions and sequences

Remember that this is called the first difference.

can be shown as a

Tip You can think of a function as ‘the

utput for a given rule.

Once you have found the first difference you can compare it with number patterns you already know to find the rule for the sequence. 1 Find the next three terms in each of these sequences. Find Explain an expression forfound the nth term of the sequence: 5, 8, 11, 14, 17 … how you them.

exercISe 4A

add 3 to it to get a

Problem-solving answer you will get’ if you scaffolding apply this rule to a number.

Instructional steps to assist students working through problem-solving 9 is reduced questions. The scaffolding as a topic progresses, so that students build up the skills and techniques required to answer further problemsolving questions without assistance.

a 4, 7, 10, 13, … Steps for approaching a problem-solving c 27, 23, 19, … question e 1, 2, 4, 8, …

Step 1: Identify what you g 4, 12, 36, … have to do.

Walk through finding the nth term on GCSE Mathematics Online.

b 38, 43, 48, 53, … What you would do for this example d 63, 57, 51, … f 64, 32, 16, … You are trying to find an expression to work out the value of any term in the sequence. h 729, 243, 81, …

2 rule foraeach these sequences. StepGive 2: If the it isterm-to-term useful to Draw tableofshowing the position and the term:

haveaa 7, table draw 14, 21, 28,one. … c 2, 8, 32, 128, …

n T(n)

b 19, 15, 11, 7, …

4 14

d5 84, 42,821, … 11

5 17

3 the working term-to-term forthe each of these StepFind 3: Start on rule Label table with sequences. the difference between each term:

Cross-topic questions Questions covering a range of topics across the GCSE Mathematics curriculum, to act as revision and promote synoptic learning.

the problem using what Use it to generate the nextn three terms 1 in each 2 sequence. 3 4 you know. a 3.5, 5.5, 7.5, … b5 1.2, 2.4, T(n) 8 4.8, …11 14 c 1_12 , 3, 4_12 , … e 72, 36, 18, …

Step 4: Connect to other sequences and compare.

d 8, 5, 2, … 13

Tip

5 17

It can help to write out the sequence and label the difference between each term.

f 210, 27, sequence 24, … T he difference in this is ‘1 3’.

Another sequence that has the same term-to-term rule of ‘add 3’ is the multiples of 3.

49 (If the difference was ‘12’ you would compare to 2n; if it was ‘24’ you would compare it to 24n.) Add this sequence to your table: n T(n) Multiples of 3n

Calculator tip

1 5 3

2 8 6

3 11 9

4 14 12

5 17 15

Compare the multiples of 3 to the terms of the original sequence.

Helpful advice to improve students’ calculator skills.

Each term is the corresponding multiple of 3 with 2 added. So the expression for the nth term of this sequence could be 3n 1 2. Step 5: Check your working and that your answer is reasonable.

Test for n 5 5: (3 3 5) 1 2 5 15 1 2 15 1 2 5 17 The fifth term is 17 so the expression is correct.

Step 6: Have you answered the question?

Yes. The expression for the nth term is 3n 1 2.

Tip If you are struggling to find the rule for an arithmetic sequence, you can use the following formula: (a 1 d(n 2 1)) where a 5 first term, and d 5 common difference. The common difference is the constant difference between terms.

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tics (Foundation)

Did you know?

did you know?

re about the Fibonacci SE Mathematics

Interesting facts to relate topics to the real world and stimulate students.

The Fibonacci pattern is found in many natural situations. In the Fibonacci series, the sequence of numbers is created by adding the 1st and 2nd term together to make the 3rd term; adding the 2nd and 3rd terms together to make the 4th term and so on.

13 branches 8 branches 5 branches 3 branches 2 branches 1 branch 1 branch

exercISe 4d 1

a Write down the sequence of the first 10 square numbers. b How could you use this sequence to find the next 2 square numbers?

a What type of number is shown here? Explain how you know this.

Work it out Questions or problems are presented with three possible solutions to choose from. The incorrect solutions are based on common errors that can be used as teaching points.

GCSE Mathematics (Foundation)

b T he picture represents the fifth term in the sequence.

Checklist of learning and understanding Find the first and second differences between the terms of the

checklist of learning and understanding

Write down the first ten numbers in the same sequence.

sequence.

A summary of the chapter content, against which students can check their knowledge and understanding in Honeybees live in colonies known as hives. order to identify any gaps.

Sequences Sequences can be formed using a term-to-term rule. Each term is generated by applying the same rule to the previous term.

d What type of sequence is this?

T here is one queen bee, a female, who is the only one able to lay eggs.

All other female bees are called workers, they are made when a male fertilises the queen’s eggs. T his means that worker bees have a mother and a father. T he males are called drones, and are made when the queen’s eggs hatch without being fertilised by a male. T his means that a drone has a mother but not a father.

T he position-to-term rule is used to find the value of any term, known as the nth term in a sequence using its position in the sequence. Functions A function is an expression or rule for changing one number (the input) into another number (the output). A sequence can be generated by inputting an ordered set of numbers into a function. Special sequences It is important to be able to recognise familiar sequences such as square numbers, cube numbers, triangular numbers and Fibonacci numbers. In a quadratic sequence the first difference between the terms is not constant but the second difference is, and the pattern is linked to square numbers.

Chapter review Each chapter ends with a review exercise that can be used to evaluate students’ understanding of the topics covered.

chapter review 4 1

For each sequence: a find the missing terms. b write an expression in terms of n to find any term in the sequence. c use your expression to find the 25th term in each sequence. i 1.5, 2, ii

iii _12 , _14 ,

Answers Each Student Book contains answers (without working) to all exercise questions to enable peer marking.

iv 5, 2

, 3, 3.5,

, 28, 24,

,…

, 4, 8,

1 __ , __ , 1, 16 32

, 17, 23, 29,

,…

,… ,…

T his array of numbers is called Pascal’s triangle.

1 1

Each number is the sum of the two numbers above it, except for the edges which are all 1. T here are many different number patterns in the triangle.

1 1 1 1

3 4

1 2

1 3

1 6 4 1 10 10 5 1

T he first row (containing the number 1) is the 0th row. a Copy and complete Pascal’s triangle to the 10th row. b Find the total for each row.

Describe the sequence that is created by these totals. Edexcel_Mathemetics.indd 11

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begins in the second row 1, 3, 6, 10, …?

Blended digital and print resources specifically created for the new AQA GCSE Mathematics specification, available from early 2015.

Brighter Thinking

For more information or to speak to your local sales consultant, please contact us:

www.cambridge.org/ukschools ukschools@cambridge.org 01223 325 588 CUPUKschools

MATHEMATICS Foundation Student Book Sample

Written from draft specification

15 Vectors In this chapter you will learn how to … • represent vectors as a diagram or column vector. • add and subtract vectors. • multiply vectors by a scalar.

Using mathematics: real-life applications Vectors have huge applications in the physical world. For example, mathematical modelling of objects sliding down slopes with varying amounts of friction, working out how far objects can tilt before they tip over and making sure two ships don’t crash in the night. All these problems involve the use of vectors.

‘When landing at any airport I have to consider how the wind will blow me off course. Over a set amount of time I expect to travel through a particular vector but I have to add on the effect the wind has on my flight path. If I don’t do this accurately I would struggle to land the plane safely.’ (Pilot)

Before you start … You need to be able to plot coordinates in all four quadrants.

1 Draw a set of axes going from 26 to 6 in both directions.

KS3

You need to be able to add, subtract and multiply negative numbers.

2 Calculate. a 3 2 7 b 24 1 11 c 25 2 18 d 24 3 7 e 23 3 29

KS3 Ch8

You need to be able to solve simple linear equations.

3 Solve. a 12 5 4m 2 36 b 2k 1 15 5 7 c 26 1 5d 5 241

KS3 Ch 8

You need to be able to solve simultaneous linear equations.

4 Solve.

KS3

450

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Plot the points A(2, 3), B(23, 4) and C(22, 23).

3x 1 2y 5 8 and 4x – 3y 5 5 © Cambridge University Press 2014 Written from draft specification

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15 Vectors

Assess your starting point using the Launchpad Step 1 ⟶ 1 Give the column vector for  HG.  H

Go to Section 1: Vector notation and representation

⟶ ⟶ 3 2 2 Draw the triangle ABC where  AB  5 (  ) and CA     5 ( )  . 25 7

✓ Step 2 21 24 2 3 j 5 (  ) k 5 ( )  l 5 ( )  3 1 22 Write the following as single vectors. a j 1 k b 2k 2 l

4 Find the values of f and g. 22 10 f ( g)   − 4 (  )  5 ( ) . 18 23 B

5 In the diagram below ⟶ ⟶ 14 9     5 (  ) .  AC  5 (  )and AB  2 12

Go to Section 2: Vector arithmetic

Find. ⟶ a  CA  ⟶ ⟶ b CA     1  AB  C

6 Which of these vectors are parallel? 15 23 9 23     ( )  (   )  (   4 16 220 ) ( 2)

✓ Go to Section 3: Mixed practice

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GCSE Mathematics (Foundation)

Section 1: Vector notation and representation Key vocabulary vector: a quantity that has both magnitude and direction. displacement: a change in position.

A vector describes movement from one point to another, it has a direction and a magnitude (size). Vectors can be used to describe many different kinds of movement. For example: displacement of a shape following translation, displacement of a boat during its journey, the velocity of an object, and the acceleration of an object. A vector that describes the movement from A to B can be represented by: an arrow in a diagram B a

⟶ AB    (arrow indicates direction) a (if handwritten this would be underlined, a) x a column vector ( y) 

Tip You include the negative sign because we are now moving in the opposite direction.

If you were to travel along this vector in the opposite direction, from B to A, you would represent this vector as: ⟶ BA     –a

2 ( 2  ) 24

Column vectors In a column vector x represents the horizontal movement; y represents the vertical movement. Movement x y positive right up negative left down ⟶ 2 In the diagram,  AB ( )  . 4 B a

452

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15 Vectors

Exercise 15A 1 Match up equivalent representations of the vectors. 1

3 H

E E

A (

24   22)

B ( ) 

C (

24)

4 2

⟶

i FE  

⟶

ii HG  

iii EF  

D ( ) 

3 0

⟶

iv GH  

E (

24)

⟶

v FE  

2 Use the diagram to answer the following questions.

C B

D F H G

Find: ⟶ a AB     ⟶ d  DF

⟶

b  DC

c  BC

e HF  

f BH  

⟶

g What do you notice about  AB and DC   ?

⟶

h What do you notice about  AB and BH   ?  3 Draw a pair of axes where x and y vary from 28 to 8.

Plot the point A (2, 21). T hen plot points B, C, D, E, F and G where: ⟶ 2 AB     5 (  ) 7

⟶ 23 AC     5 (  ) 7

⟶ 26  AD  5 (  ) 3

⟶ 5 AE     5 (  ) 3

⟶ 23 AF     5 (  ) 21

⟶ 2  AG  5 (   ) 21

4 a Find the vector from point A with coordinates (3, 24) to point B with

coordinates (21, 2). b Give the coordinates of two more points E and F where the vector from

⟶ E to F is the same as  AB.

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GCSE Mathematics (Foundation)

Tip T he midpoint of the line KL is halfway the along the line from K to L.

5 a Find the vector from point K with coordinates (22, 21) to point L with

coordinates (28, 9). b Use your answer to find the coordinates of the midpoint of KL. 6 T hese vectors describe how to move between points A, B, C, and D.

⟶ ⟶ ⟶ 1 21 2 BC     5 (  ) DA     5 (  )  AB  5 (  ) 0 2 1 Draw a diagram showing how the points are positioned to form the quadrilateral ABCD.

7 T he vector (

28)

  describes the displacement from point A to point B.

a What is the vector from point B to point A? b Point A has coordinates (3, 5). What are the coordinates of point B?

Section 2: Vector arithmetic Addition and subtraction ⟶ ⟶ 4 6 2 ⟶ T he diagram shows AB     5 ( )  , BC     5 (  )  , and AC     5 (  ) 22 4 2

C A

Moving from A to B and then from B to C is the same as moving directly from A to C. In other words, you can take a ‘shortcut’ from A to C by adding ⟶ ⟶   .  together AB    and BC ⟶ ⟶ ⟶    and BC   .   AC is known as the resultant of AB ⟶ ⟶ ⟶    5 AC      AB 1 BC 4 6 2 ( 4 ) 1 ( 22 ) 5 (2 ) ⟶ ⟶ T he diagram shows AB    and CB   .  ⟶ ⟶     To find AC    you need to travel along CB in the opposite direction. ⟶ So, subtract  CB.  ⟶ ⟶ ⟶    5 AC      AB – CB 24 − 2 2 26  5    (  ) − (  ) 5 ( 5 1 1 − 5) (24) 24

454

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B A

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15 Vectors

Multiplying by a scalar Multiplying a vector by a scalar results in repeated addition.

Key vocabulary

T his is the same as multiplying the x-component by the scalar, k, and the y-component by the same scalar, k.

scalar: a numerical quantity (it has no direction).

⟶ ⟶ 4 12 AB    5 (    ).    5 (    )and CD 21 23 ⟶     CD 5 3AB A B

B A

A B

repeated addition of AB

4 4 4 4 repeated addition 3 3 (   )5 (    )1 (    )1 (   ) 21 21 21 21 multiplying the x-component by the scalar k 3 3 4   5 ( 3 3 21) multiplying the y-component by the scalar k

12 5  (  )  23

Multiplying a vector by a scalar k results in a parallel vector with a magnitude multiplied by k.

Key vocabulary parallel vectors: occur when one vector is a multiple of the other.

Vectors are parallel if one is a multiple of the other. Work it out 15.1

Which of the following vectors are parallel? a 5 (

21)

b 5 (

c 5 (

23)

6 d 5 (  ) 2

e 5 (

Option A

Option B

Option C

Vectors a and c

Vectors d and e

Vectors a, c and e

Exercise 15B 1 p 5 (

5 q 5 (    ) 21

23 r 5 (  ) 22

4 s 5 (   )  27

Write each of these as a single vector. a p 1 q

b s − r

c 4p

d 23s

e p 1 q 1 r

f 2p 1 q − 2s

g Which of the results from parts a to f are parallel to the vector (

22)

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2 Give three vectors parallel to (

23)

2   .

3 Find x, y, z and t in each of the following vector calculations. b ( y)   − ( 23)   5 ( )  8

a (  3)   1 (y)   5 ( )  3

c ( 23 ) 1 ( y ) 5 (  )  28

e z(

d z( 12 ) 5 (

224)

f (   )   1 z(y)   5 ( )  24 14

3   )  y)   5 (2 8

212

g ( 24 ) − z(

25 20    5 (  ) 5 23)

h z(  ) 1 t(

18    5 (  ) 10

22)

3 4

4 In the diagram,  AB  5 ( )  .

⟶

20 16 T he ratio of AC : CB is 1 : 3. ⟶ a Find AC   .  ⟶ b Find  BC.

Tip You can multiply a vector by a fractional scalar if you need to divide. If you need a reminder on fractions see chapter 3; if you need a reminder of how to calculate ratios, see chapter 10.

C A

5 T hese vectors describe how to move between points E, F, G and H,

which are four vertices of a quadrilateral. ⟶ ⟶ ⟶ 3 6 0     5 (   ) EH     5 (  )  EF  5 (   ) HG 21 22 1 a What can you say about sides EF and HG? b Predict what kind of quadrilateral EFGH is.

⟶

c Draw the quadrilateral and find  GF.

⟶

6 ABCD is a quadrilateral.  AB 5 DC    and DA    5 CB   .

What kind of quadrilateral is ABCD? How do you know this?

Section 3: Mixed practice It is important that you understand what calculations are needed when given a problem involving vectors. Test your knowledge using the exercise.

Exercise 15C Tip Remember that the line CM is half the length of the line C B. What does this mean about the journey from C to M?

⟶ ⟶ 10 8 1 In the diagram,  AC  5 (  )and AB     5 (  )  . 14 2 M is the midpoint of BC.

Find: ⟶ a CA   .  ⟶ ⟶ b  CA  1  AB.  ⟶ c  CM

C A

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2 Two triangles have vertices ABC and DEF.

T he coordinates of the vertices are A(0, 0), B(3, 2), C(2, 5) and D(1, 1), E(7, 5), F(5, 11). Compare the vectors. ⟶ ⟶ a AB    and DE     ⟶ ⟶ b  AC and DF     c What does this tell you about the triangles ABC and DEF? 3 In a game of chess different pieces move in different ways.

A king can move one square in any direction (including diagonals). A knight moves two squares horizontally and one square vertically or two squares vertically and one horizontally. A chessboard is eight squares wide and eight squares long. What vectors can the following pieces move? a King b Knight 4 A ship travels 8 km east and 10 km north. What vector has it travelled? 5 T he vector from E to F is (

5   and the vector from F to G is (  ). 1 2)

What is the vector from: a E to G? b G to F? c E to the midpoint of EF? d G to the midpoint of EF? 6 T he vector from A to B is (

22)

T he vector joining C to D is parallel to AB, D is five times the distance from C as B is from A. What is the vector from C to D?

Checklist of learning and understanding Notation

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Multiplication by a scalar To multiply by a scalar you can use repeated addition, or multiply the x-component by the scalar and the y-component by the scalar. 26 22 22 22 22 3(  ) 5 (  ) 1 (  ) 1 (  ) 5 (  ) 1 1 1 1 3 26 22 3(  ) 5 ( )  1 3 Multiplying a vector by a scalar quantity produces a parallel vector; you can identify that vectors are parallel if one vector is a multiple of the other. Parallel vectors can be part of the same line and described using a ratio.

Chapter review 15 1 What is the difference between coordinate (22, 3) and vector (

  ? 3)

2 Match the parallel vectors. 26 a 5 (  ) 2 22 e 5 (  ) 4

1 b 5 (  ) 3 26 f 5 (  ) 12

3 Calculate. a (

22)

22 1    1 (  ) 21

b (

3 c 5 (    ) 21 21 g 5 (  ) 2

23)

7 d 5 (   ) 21

c 23(

21)

0 22    2 (  ) 4

⟶ ⟶

4 T he ratio of  AB : BC    is 2 : 3.

⟶ 26 If  AB 5 (  )what is the column vector of 4 ⟶ ⟶ a BC   ?  b  AC?

C A

5 In the diagram, M is the midpoint of AB.

Find ⟶ a AB     ⟶ b  AM  ⟶ c  MO

a M

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17 Vectors In this chapter you will learn how to … • represent vectors as a diagram or column vector. • add and subtract vectors. • multiply vectors by a scalar. • use vectors to construct geometric arguments and proofs.

1 Draw a set of axes going from 26 to 6 in both directions.

KS3

You need to be able to add, subtract and multiply negative numbers.

2 Calculate. a 3 2 7 b 24 1 11 c 25 2 18 d 24 3 7 e 23 3 29

KS3 Ch8

You need to be able to solve simple linear equations.

3 Solve. a 12 5 4m 2 36 b 2k 1 15 5 7 c 26 1 5d 5 241

KS3 Ch 8

You need to be able to solve simultaneous linear equations.

4 Solve.

KS3

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Plot the points A(2, 3), B(23, 4) and C(22, 23).

3x 1 2y 5 8 and 4x – 3y 5 5 © Cambridge University Press 2014 Written from draft specification

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17 Vectors

Assess your starting point using the Launchpad Step 1 ⟶ 1 Give the column vector for  HG.

2 Draw the triangle ABC where

⟶ ⟶ 3 2     5 ( )  .  AB  5 (  ) and CA 25 7

Go to Section 1: Vector notation and representation

✓ Step 2 21 24 2 3 j 5 ( )  k 5 ( )  l 5 ( )  3 1 22 Write the following as single vectors. a j 1 k b 2k 2 l

4 Find the values of f and g. 22 10 f ( g)   − 4 (   ) 5 ( ) . 18 23

5 In the diagram below ⟶ ⟶ 14 9     5 (  ) .  AC  5 (  )and AB  2 12 Find. ⟶ ⟶ ⟶ a  CA   b CA     1  AB

6 Which of these vectors are parallel? 15 23 9 23     ( )  (   )  (   4 16 220 ) ( 2)

Go to Section 2: Vector arithmetic

✓ Step 3 7 ABCD is a square. ⟶ ⟶    5 b  AB 5 a, BC

If the ratio of AB : AE is 1 : 2, find ⟶ ⟶ a BE     b  AF  M is the midpoint of EF. ⟶ c Find: AM  

b D

✓

M F

Go to Section 3: Using vectors in geometric proofs

Go to Section 3: Chapter review 33

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Section 1: Vector notation and representation Key vocabulary vector: a quantity that has both magnitude and direction. displacement: a change in position.

⟶ AB    (arrow indicates direction) a (if handwritten this would be underlined, a) x a column vector ( y) 

Tip You include the negative sign because you are now moving in the opposite direction.

If you were to travel along this vector in the opposite direction, from B to A, you would represent this vector as: ⟶ BA     –a

2 ( 2  ) 24

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Exercise 17A 1 Match up equivalent representations of the vectors. 1

3 H

E E

A (

24   22)

B ( ) 

C (

24)

4 2

⟶

i FE  

⟶

ii HG  

iii EF  

D ( ) 

3 0

⟶

iv GH  

E (

24)

⟶

v FE  

2 Use the diagram to answer the following questions.

C B

D F H G

Find each of the vectors: ⟶ ⟶ a AB     b  DC   ⟶ ⟶ d  DF   e HF  

⟶

c  BC

⟶

f BH  

⟶

g What do you notice about  AB and DC   ?

⟶

h What do you notice about  AB and BH   ?  3 Draw a pair of axes where x and y vary from 28 to 8.

Plot the point A (2, 21). T hen plot points B, C, D, E, F and G where: ⟶ 2 AB     5 (  ) 7

⟶ 23 AC     5 (  ) 7

⟶ 26  AD  5 (  ) 3

⟶ 5 AE     5 (  ) 3

⟶ 23 AF     5 (  ) 21

⟶ 2  AG  5 (   ) 21

4 a Find the vector from point A with coordinates (3, 24) to point B with

coordinates (21, 2). b Give the coordinates of two more points E and F where the vector from

⟶ E to F is the same as  AB.

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Tip T he midpoint of the line KL is halfway the along the line from K to L.

5 a Find the vector from point K with coordinates (22, 21) to point L

with coordinates (28, 9). b Use your answer to find the coordinates of the midpoint of KL. 6 T hese vectors describe how to move between points A, B, C, and D.

⟶ ⟶ 1 21 BC     5 (  ) DA     5 (  ) 0 2 Draw a diagram showing how the points are positioned to form the quadrilateral ABCD.

⟶ 2  AB  5 (  ) 1

7 In a game of chess, different pieces move in different ways.

A king can move one square in any direction (including diagonals). A bishop can move any number of squares diagonally. A knight moves two squares horizontally and one square vertically or two squares vertically and one horizontally. A chessboard is eight squares wide and eight squares long. What vectors can the following pieces move? a Bishop

b King

c Knight

For part a, you will need to think algebraically 8 How would you find the length (magnitude) of a vector?

How could you describe its direction? Use these diagrams and your knowledge of Pythagoras and trigonometry to help design a method. 3 2 Vector (  ) Vector (    ) 24 5

Tip See chapter 14 on Pythagoras’ theorem and chapter 15 on trigonometry if you need a reminder.

Section 2: Vector arithmetic Addition and subtraction ⟶ ⟶ 4 6 2 ⟶ T he diagram shows AB     5 (  ), BC     5 (   ), and AC     5 (  ) 22 4 2 B

C A

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Moving from A to B and then from B to C is the same as moving directly from A to C. In other words, you can take a ‘shortcut’ from A to C by adding ⟶ ⟶   .  together AB    and BC ⟶ ⟶ ⟶    and BC   .   AC is known as the resultant of AB ⟶ ⟶ ⟶    5 AC      AB 1 BC 4 6 2 ( 4)   1 ( 22)   5 (2)  ⟶ ⟶   .  T he diagram shows AB    and CB ⟶ ⟶        you need to travel along CB To find AC in the opposite direction. ⟶ So, subtract  CB.

B A

⟶ ⟶ ⟶    5 AC      AB – CB

24 − 2 2 26  5    (  ) − (  ) 5 ( 5 1 1 − 5) (24) 24

Multiplying by a scalar Multiplying a vector by a scalar results in repeated addition. T his is the same as multiplying the x-component by the scalar, k, and the y-component by the same scalar, k. ⟶ ___ ⟶ ___ 4 12 AB    5 (     ) .    5 (       ) and CD 21 23 ⟶     CD 5 3AB

Key vocabulary scalar: a numerical quantity (it has no direction).

A C

B B A

A B

repeated addition of AB

4 4 4 4 3 3 (   )5 (    )1 (    )1 (   ) 21 21 21 21 3 3 4   5 ( 3 3 21) 12 5  (  )  23

repeated addition

multiplying the x-component by the scalar k multiplying the y-component by the scalar k

Key vocabulary

Multiplying a vector by a scalar k results in a parallel vector with a magnitude multiplied by k. Vectors are parallel if one is a multiple of the other. © Cambridge University Press 2014 Written from draft specification

Edexcel Chapter 17 Higher.indd 485

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Work it out 17.1

Which of the following vectors are parallel? a 5 (

21)

b 5 (

c 5 (

23)

6 d 5 (  ) 2

e 5 (

Option A

Option B

Option C

Vectors a and c

Vectors d and e

Vectors a, c and e

Exercise 17B 1 p 5 (

23 r 5 (  ) 22

5 q 5 (    ) 21

4 s 5 (    ) 27

Write each of these as a single vector. a p 1 q

b s − r

c 4p

d 23s

e p 1 q 1 r

f 2p 1 q − 2s

g Which of the results from parts a to f are parallel to the vector (

2 Give three vectors parallel to (

23)

22)

3 Find x, y, z and t in each of the following vector calculations. b ( y  ) − ( 23 ) 5 (

a (  3 ) 1 (y ) 5 (  )

c ( 23 ) 1 ( y ) 5 (   ) 28

9 3

d z( 12  )  5 (    ) 224

e z(

3 y)   5 ( 28)  x 25 20 g ( 24 ) − z(  ) 5 (   ) 23 5

f (   )   1 z(y )  5 ( )  24 14

212

h z (  ) 1 t(

18    5 (  ) 10

22)

3 4

4 In the diagram,  AB  5 (  ).

⟶

20 16 T he ratio of AC : CB is 1 : 3. ⟶ a Find AC   .  ⟶ b Find  BC.

Tip You can multiply a vector by a fractional scalar if you need to divide. If you need a reminder on fractions see chapter 3; if you need a reminder of how to calculate ratios, see chapter 10.

C A

5 T hese vectors describe how to move between points E, F, G and H,

which are four vertices of a quadrilateral. ⟶ ⟶ ⟶ 3 6 0     5 (    ) EH     5 (  )  EF  5 (   ) HG 21 22 1 a What can you say about sides EF and HG? b Predict what kind of quadrilateral EFGH is.

⟶

c Draw the quadrilateral and find  GF.

⟶

6 ABCD is a quadrilateral.  AB 5 DC    and DA    5 CB   .

What kind of quadrilateral is ABCD? How do you know this?

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Section 3: Using vectors in geometric proofs Vectors can be used to prove geometric results. You can use them to identify parallel lines find midpoints share lines in a given ratio. ⟶ For example, in this triangle AB    5 2a 1 b 5b2a A M a B

N b O

M is the midpoint of AB. T he ratio of ON : NA is 1: 2 ⟶ You can use what you know about vectors to find AM   .  ⟶ You know that AM is half of AB, so it follows that AM    is half of the journey from A to B, so ⟶ _1  AM  5  2 (b 2 a) You can use what you know about the effect of scalars on vectors to ⟶ calculate ON   .  You know that the ratio of ON : NA is 1: 2.

⟶ ⟶ So, you know that the point N is such that 2ON     5  NA,  so ⟶ _1 ON     5  3 a Worked example 1

In this parallelogram, M is the midpoint of AC, and N is the midpoint of BC. ⟶ ⟶   .  Prove that MN    is parallel to AB M

a N O

⟶    5 b 2 a AB ⟶ ⟶ ⟶ ⟶ ⟶     1  AO  1  OB  1  BN   MN 5 MA

Tip If you’re struggling with a question, highlight sides that are labelled with vectors or that are parallel to any given vectors. T hen identify the journey between your two points by using these highlighted lines.

 2 b 2 a 1 b 1 _  2 a 5 2_ 1

5_  2 (b 2 a) ⟶ ⟶   , they are parallel. Since MN    is a multiple of AB 1

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Worked example 2

⟶ → In the regular hexagon shown, EF    5 e and  JI    5   j.

It is possible to move between any two vertices of the regular hexagon using combinations of vectors e and j.

e E

Find the vector that describes each journey. ⟶ → → a EG     b  HJ   c  EJ

Opposite sides of a regular hexagon are parallel.

e E

Tip

G j

j e

Drawing a few additional lines parallel to the vectors given can help you to see a solution.

j I

⟶ → ⟶ a EG     =  EF  +  FG  ⟶  FG  = j ⟶ EG    = –e + j ⟶ → → b  HJ  =  HI  +  IJ

Write these in terms of e and j.

⟶ Using addition of vectors, EG    is the resultant vector. Using the properties of a regular hexagon, the line FG is parallel to JI. Parallel vectors are a multiple of each other, in this case the scalar is 1.

→ Using addition of vectors, HJ    is the resultant vector.

→  HI  = − e

Using the properties of a regular hexagon, the line IH is parallel to EF. Parallel vectors are a multiple of each other, in this case the scalar is 1. You are travelling from H to I, in the opposite direction of e, so need the negative of e.

⟶ HJ     = −e−j

→ →  IJis   the opposite direction of  JI .

⟶ ⟶ ⟶ ⟶⟶ c  E J = EH    +  H I  +  IJ   EH = 2b

→ Using addition of vectors, EJ    is the resultant vector.

⟶ EH    = 2j

Using the helpful additional lines, EH is parallel to IJ and twice its length. It is moving in the same direction.

→ H   I 5 2e

→ →    is the opposite direction to IH HI   .

→ IJ   2j   5 ⟶   = EH   2j 2 e 2 j = j – e

→ →  IJ  is the opposite direction to  IJ .

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17 Vectors

Exercise 17C 1 In the diagram,  AC  5 (   )and AB     5 (    ). 2 14

⟶

C A

M is the midpoint of BC. Find: ⟶ a CA     ⟶ ⟶ b  CA  1  AB   ⟶ c  CM  2 Two triangles have vertices ABC and DEF:

A(0,0), B(3,2), C(2, 5) D(1,1), E(7,5), F(5,11). Compare the vectors. ⟶ ⟶ a AB    and DE     ⟶ ⟶ b  AC and DF     c What does this tell you about the triangles ABC and DEF? 3 MNOP is a square.

Find the vectors. Explain your answers. ⟶ ⟶ a NO     b  OP  ⟶ ⟶ c  MO   d  PN

4 ABCD is a parallelogram.

M is the midpoint of side BC, N the midpoint of CD. ⟶ ⟶    5 q.  AB 5 p and BM q

p A

Find the vectors. Explain your answers. ⟶ ⟶ a AC     b  DB  ⟶ ⟶ ⟶ c  MD   d Show that  NM is parallel to DB   .  © Cambridge University Press 2014 Written from draft specification

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5 T he diagram shows four congruent triangles forming a tessellating pattern,

Tip Congruent shapes are exactly the same shape and size. You will learn more about congruency in chapter 16.

and vectors m and n. F

m J

Find the vectors → → a  IJ      b  EJ

⟶

d  EH

c   JF 

6 EFG is an equilateral triangle.

Points H, I and J are the midpoints of each side. F

⟶ ⟶ EF    5 g.    5 e and EG Find the vectors ⟶ a EH     → d  HI

⟶

→

b  JE

c  FG

→

e  IJ   

What can you say about triangle HIJ? 7 EFG is an equilateral triangle.

Points H and I are the midpoints of sides EF and FG. JI is a straight line with midpoint H. ⟶ ⟶    5 g.  EF 5 e and EG F

Find the vectors → ⟶ a FH     b  IG

→

c  HI

d   JI 

8 In the diagram, V is the midpoint

of TR and W the midpoint of RS. T he ratio of TU : US is 1 : 4. ⟶ ⟶    5 T.  SR 5 R and TU Find the vectors → ⟶ a TS     b  UR   ⟶ ⟶ c  VR   d  WV 

490

Edexcel Chapter 17 Higher.indd 490

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e JE  

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17 Vectors

9 In the diagram, the ratio of PM : MQ is 3 : 1. Q M

q P p

⟶ a Find PQ     ⟶ b Find  PM   ⟶ 1 c Show  OM 5 _ 4 ( 3q 1 p)

10 A river runs from east to west at 3 m every second.

It is 12 m wide. James can swim at 1.5 m every second. He sets off to cross the river. How far off course is he when he reaches the other river bank? How far does he actually swim?

Checklist of learning and understanding Notation

⟶ 1 Vectors can be written in a variety of ways: AB   ,  a, (  ). 2 Addition and subtraction To add or subtract vectors simply add or subtract the x- and y-components. 5 26 5 3 2 21   )   5 ( )  ( 2)   1 ( 24)   5 ( 22)  ( 4)   2 (2 6 10 Multiplication by a scalar To multiply by a scalar you can use repeated addition; or multiply the x-component by the scalar and the y-component by the scalar. 26 22 22 22 22 3(  ) 5 (  ) 1 (  ) 1 (  ) 5 (  ) 1 1 1 1 3 26 22 3(  ) 5 (  ) 1 3 Multiplying a vector by a scalar quantity produces a parallel vector; you can identify that vectors are parallel if one vector is a multiple of the other. Parallel vectors can be part of the same line and described using a ratio.

Using vectors in geometric proofs You can use vectors to identify parallel lines, find midpoints and share lines in a given ratio. © Cambridge University Press 2014 Written from draft specification

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Chapter review 17 1 What is the difference between coordinate (22, 3) and vector (

  ? 3)

2 Match the parallel vectors. 26 a 5 (  ) 2 22 e 5 ( )  4

1 b 5 (  ) 3 26 f 5 ( )  12

3 Calculate.

b (

23)

a (   )   1 ( )  22 21

3 c 5 (    ) 21 21 g 5 ( )  2

7 d 5 (    ) 21

c 23(

21)

22    2 (  ) 4

4 EFG is a straight line.

EF : FG 5 2 : 3 G

13e 1 7f F O

Find. ⟶ a EG  

3e 1 2f

⟶

b  FG

⟶

c  OF

5 T he points G and H lie on line EF.

⟶  EF 5 12e 2 18f

T he ratio of EG : GH : HF is 1 : 3 : 2. a Draw a sketch of this situation. b Find the vectors

⟶

i  GH

492

Edexcel Chapter 17 Higher.indd 492

⟶

ii HE  

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