1 properties of this chapter will show you … vectors

559

This chapter will show you …

● the properties of vectors

● how to add and subtract vectors

● how to use vectors to solve geometrical problems

Visual overview

What you should already know

● Vectors are used to describe translations

Quick check

Use column vectors to describe these translations.

a A to C

b B to D

c C to D

d D to E

C

B D

A E

1 Properties ofvectors

2 Vectors ingeometry

Properties

Addition and subtraction Solving geometrical problems

Vectors

Edex_Higher Math_25.qxd 16/03/06 13:13 Page 559

© HarperCollinsPublishers Limited 2007

A vector is a quantity which has both magnitude and direction. It can be represented by a straight linewhich is drawn in the direction of the vector and whose length represents the magnitude of the vector.Usually, the line includes an arrowhead.

The translation or movement from A to B is represented by thevector a.

a is always printed in bold type, but is written as a.

a can also be written as AB⎯→

.

A quantity which is completely described by its magnitude, and has no direction associated with it, is called a scalar. The mass of a bus (10 tonnes) is an example of a scalar. Another example is a linearmeasure, such as 25.4 mm.

Multiplying a vector by a number (scalar) alters its magnitude (length) but not its direction. For example,the vector 2a is twice as long as the vector a, but in the same direction.

A negative vector, for example –b, has the same magnitude as the vector b, but is in the oppositedirection.

Addition and subtraction of vectors

Take two non-parallel vectors a and b, then a + b is defined to be the translation of a followed by thetranslation of b. This can easily be seen on a vector diagram.

ab b

a

a + b

b –b

a 2a

a

A

B

560

Properties of vectors25.1

Key wordsdirectionmagnitudevector

In this section you will learn how to:● add and subtract vectors



Similarly, a – b is defined to be the translation of a followed by the translation of –b.

Look at the parallelogram grid below. a and b are two independent vectors that form the basis of this grid.It is possible to define the position, with reference to O, of any point on this grid by a vector expressed interms of a and b. Such a vector is called a position vector.

For example, the position vector of K is⎯→OK or k = 3a + b, the position vector of E is

⎯→OE or e = 2b.

The vector ⎯→HT = 3a + b, the vector

⎯→PN = a – b, the vector

⎯→MK = 2a – 2b, and the vector

⎯→TP = –a – b.

Note ⎯→OK and

⎯→HT are called equal vectors because they have exactly the same length and are in the same

direction.⎯→MK and

⎯→PN are parallel vectors but

⎯→MK is twice the magnitude of

⎯→PN.

b

a A C F J

KGD

O

B

E H L P R

N

S TQMI

a a

a – b

–b–b

561

CHAPTER 25: VECTORS

EXAMPLE 1

a Using the grid above, write down the following vectors in terms of a and b.

i⎯→BH ii

⎯→HP iii

⎯→GT

iv→T I v

⎯→FH vi

⎯→BQ

b What is the relationship between the following vectors?

i⎯→BH and

⎯→GT ii

⎯→BQ and

⎯→GT iii

⎯→HP and

→T I

c Show that B, H and Q lie on the same straight line.

a i a + b ii 2a iii 2a + 2b iv – 4a v –2a + 2b vi 2a + 2b

b i⎯→BH and

⎯→GT are parallel and

⎯→GT is twice the length of

⎯→BH.

ii⎯→BQ and

⎯→GT are equal.

iii⎯→HP and

→TI are in opposite directions and

→T I is twice the length of

⎯→HP.

c⎯→BH and

⎯→BQ are parallel and start at the same point B. Therefore, B, H and Q must lie on

the same straight line.



On this grid,⎯→OA is a and

⎯→OB is b.

a Name three other vectors equivalent to a.

b Name three other vectors equivalent to b.

c Name three vectors equivalent to –a.

d Name three vectors equivalent to –b.

Using the same grid as in question 1, give the following vectors in terms of a and b.

a⎯→OC b

⎯→OE c

⎯→OD d

⎯→OG e

⎯→OJ

f⎯→OH g

⎯→AG h

⎯→AK i

⎯→BK j

→DI

k⎯→GJ l

⎯→DK

a What do the answers to parts 2c and 2g tell you about the vectors⎯→OD and

⎯→AG?

b On the grid in question 1, there are three vectors equivalent to⎯→OG. Name all three.

a What do the answers to parts 2c and 2e tell you about vectors⎯→OD and

⎯→OJ?

b On the grid in question 1, there is one other vector that is twice the size of⎯→OD. Which is it?

c On the grid in question 1, there are three vectors that are three times the size of⎯→OA.

Name all three.

a

b

A C

D G

E H J K

I

O F

B

562

CHAPTER 25: VECTORS

EXAMPLE 2

Use a vector diagram to show that a + b = b + a

Take two independent vectors a and b

a + b and b + a have the same magnitude and direction and are therefore equal.

bb

a

a

a + bb + a

a b

EXERCISE 25A



On a copy of this grid, mark on the points C to Pto show the following.

a⎯→OC = 2a + 3b b

⎯→OD = 2a + b

c⎯→OE = a + 2b d

⎯→OF = 3b

e⎯→OG = 4a f

⎯→OH = 4a + 2b

g⎯→OI = 3a + 3b h

⎯→OJ = a + b

i⎯→OK = 2a + 2b j

⎯→OM = 2a + 3–2b k

⎯→ON = 1–2a + 2b l

⎯→OP = 5–2a + 3–2b

a Look at the diagram in question 5. What can you say about the points O, J, K and I?

b How could you tell this by looking at the vectors for parts 5g, 5h and 5i?

c There is another point on the same straight line as O and D. Which is it?

d Copy and complete these statements and then mark the appropriate points on the diagram youdrew for question 5.

i The point Q is on the straight line ODH. The vector⎯→OQ is given by

⎯→OQ = a + …… b

ii The point R is on the straight line ODH. The vector⎯→OR is given by

⎯→OR = 3a + …… b

e Copy and complete the following statement.

Any point on the line ODH has a vector na + …… b, where n is any number.

On this grid,⎯→OA is a and

⎯→OB is b.

Give the following vectors in terms of a and b.

a⎯→OH b

⎯→OK

c⎯→OJ d

⎯→OI

e⎯→OC f

⎯→CO

g⎯→AK h

→DI

i→JE j

⎯→AB k

⎯→CK l

⎯→DK

a What do the answers to parts 7e and 7f tell you about the vectors⎯→OC and

⎯→CO?

b On the grid in question 7, there are five other vectors opposite to⎯→OC. Name at least three.

a What do the answers to parts 7j and 7k tell you about vectors⎯→AB and

⎯→CK?

b On the grid in question 7, there are two vectors that are twice the size of⎯→AB and in the opposite

direction. Name both of them.

c On the grid in question 7, there are three vectors that are three times the size of⎯→OA and in the

opposite direction. Name all three.

a

b

I J

A F

B C D E

G

H K

O

b

a AO

B

563

CHAPTER 25: VECTORS

Edex_Higher Math_25.qxd 16/03/06 13:14 63


On a copy of this grid, mark on the points C to Pto show the following.

a⎯→OC = 2a – b b

⎯→OD = 2a + b

c⎯→OE = a – 2b d

⎯→OF = b – 2a

e⎯→OG = –a f

⎯→OH = –a – 2b

g⎯→OI = 2a – 2b h

⎯→OJ = –a + b

i⎯→OK = –a – b j

⎯→OM = –a –3–2b k

⎯→ON = –1–2a – 2b l

⎯→OP = 3–2a – 3–2b

This grid shows the vectors⎯→OA = a and

⎯→OB = b.

a Name three vectors equivalent to a + b.

b Name three vectors equivalent to a – b.

c Name three vectors equivalent to b – a.

d Name three vectors equivalent to –a – b.

e Name three vectors equivalent to 2a – b.

f Name three vectors equivalent to 2b – a.

g For each of these, name one equivalent vector.

i 3a – b ii 2(a + b) iii 3a – 2b

iv 3(a – b) v 3(b – a) vi 3(a + b)

vii –3(a + b) viii 2a + b – 3a – 2b ix 2(2a – b) – 3(a – b)

The points P, Q and R lie on a straight line. The vector⎯→PQ is 2a + b, where a and b are vectors.

Which of the following vectors could be the vector⎯→PR and which could not be the vector

⎯→PR

(two of each).

a 2a + 2b b 4a + 2b c 2a – b d –6a – 3b

The points P, Q and R lie on a straight line. The vector⎯→PQ is 3a – b, where a and b are vectors.

a Write down any other vector that could represent⎯→PR.

b How can you tell from the vector⎯→PS that S lies on the same straight line as P, Q and R?

Use the diagram in question 11 to prove the following results.

a KB is parallel to IE.

b L, A and F are on a straight line.

Use a vector diagram to show that a + (b + c) = (a + b) + c.

a

b

A

C F

J K

G

D

B

E

H

L P

O

N M

I

b

a AO

B

564

CHAPTER 25: VECTORS



Vectors can be used to prove many results in geometry, as the following examples show.

565

Vectors in geometry25.2

Key wordvector

In this section you will learn how to:● use vectors to solve geometrical problems

EXAMPLE 3

In the diagram,⎯→OA = a,

⎯→OB = b, and

⎯→BC = 1.5a. M is the midpoint of BC, N is the midpoint

of AC and P is the midpoint of OB.

a Find these vectors in terms of a and b.

i⎯→AC ii

⎯→OM iii

⎯→BN

b Prove that ⎯→PN is parallel to

⎯→OA.

a i You have to get from A to C in terms of vectors that you know.⎯→AC =

⎯→AO +

⎯→OB +

⎯→BC

Now⎯→AO = –

⎯→OA, so you can write

⎯→AC = –a + b + 3–2 a

= 1–2 a + b

Note that the letters “connect up” as we go from A to C, and that the negative of a vector represented by any pair of letters is formed by reversing the letters.

ii In the same way⎯→OM =

⎯→OB +

⎯→BM =

⎯→OB + 1–2

⎯→BC

= b + 1–2 ( 3–2 a)⎯→OM = 3–4 a + b

A

N

C

M

B

O

b

a

P



566

CHAPTER 25: VECTORS

EXAMPLE 4

OACB is a parallelogram.⎯→OA is represented by the vector a.

⎯→OB is represented by the

vector b. P is a point 2–3 the distance from O to C, and M is the midpoint of AC. Show that B, P and M lie on the same straight line.

⎯→OC =

⎯→OA +

⎯→AC = a + b

⎯→OP = 2–3

⎯→OC = 2–3 a + 2–3 b

⎯→OM =

⎯→OA +

⎯→AM =

⎯→OA + 1–2

⎯→AC = a + 1–2 b

⎯→BP =

⎯→BO +

⎯→OP = –b + 2–3 a + 2–3 b = 2–3 a – 1–3 b = 1–3 (2a – b)

⎯→BM =

⎯→BO +

⎯→OM = –b + a + 1–2 b = a – 1–2 b = 1–2(2a – b)

Therefore,⎯→BM is a multiple of

⎯→BP (

⎯→BM = 3–2

⎯→BP ).

Therefore,⎯→BP and

⎯→BM are parallel and as they have a common point, B, they must lie on the

same straight line.

B

O a

A

M

C

Pb

iii⎯→BN =

⎯→BC +

⎯→CN =

⎯→BC – 1–2

⎯→AC

= 3–2 a – 1–2 ( 1–2 a + b)

= 3–2 a – 1–4 a – 1–2 b

= 5–4 a – 1–2 b

Note that if you did this as⎯→BN =

⎯→BO +

⎯→OA +

⎯→AN, you would get the same result.

b⎯→PN =

⎯→PO +

⎯→OA +

⎯→AN

= 1–2 (–b) + a + 1–2 ( 1–2 a + b)

= – 1–2 b + a + 1–4 a + 1–2 b

= 5–4 a

⎯→PN is a multiple of a only, so must be parallel to

⎯→OA.



The diagram shows the vectors⎯→OA = a and

⎯→OB = b. M is the midpoint of AB.

a i Work out the vector⎯→AB.

ii Work out the vector⎯→AM.

iii Explain why⎯→OM =

⎯→OA +

⎯→AM.

iv Using your answers to parts ii and iii,

work out⎯→OM in terms of a and b.

b i Work out the vector⎯→BA.

ii Work out the vector⎯→BM.

iii Explain why⎯→OM =

⎯→OB +

⎯→BM.

iv Using your answers to parts ii and iii, work out⎯→OM in terms of a and b.

c Copy the diagram and show on it the vector⎯→OC which is equal to a + b.

d Describe in geometrical terms the position of M in relation to O, A, B and C.


⎯→OC = –b. N is the midpoint of AC.

a i Work out the vector⎯→AC.

ii Work out the vector⎯→AN.

iii Explain why⎯→ON =

⎯→OA +

⎯→AN.

iv Using your answers to parts ii and iii, work out⎯→ON in terms of a and b.

b i Work out the vector⎯→CA.

ii Work out the vector⎯→CN.

iii Explain why⎯→ON =

⎯→OC +

⎯→CN.

iv Using your answers to parts ii and iii, work out⎯→ON in terms of a and b.

c Copy the diagram above and show on it the vector⎯→OD which is equal to a – b.

d Describe in geometrical terms the position of N in relation to O, A, C and D.

Copy this diagram and on it draw vectors that represent

a a + b b a – b

A

B

O

b

a

N

O

C

A

–b

a

M

B

O A

b

a

567

CHAPTER 25: VECTORS

EXERCISE 25B




⎯→OB = b.

The point C divides the line AB in the ratio 1:2

(i.e. AC is 1–3 the distance from A to B).

a i Work out the vector⎯→AB.

ii Work out the vector⎯→AC.

iii Work out the vector⎯→OC in terms of a and b.

b If C now divides the line AB in the ratio 1:3 (i.e. AC is 1–4 the distance from A to B), write down

the vector that represents⎯→OC.


⎯→OB = b.

The point C divides OB in the ratio 2 :1 (i.e. OC is 2–3 the distance from O to B). The point E is such

that⎯→OE = 2

⎯→OA. D is the midpoint of AB.

a Write down (or work out) these vectors in terms of a and b.

i⎯→OC ii

⎯→OD iii

⎯→CO

b The vector⎯→CD can be written as

⎯→CD =

⎯→CO +

⎯→OD. Use this fact to work out

⎯→CD in terms of a

and b.

c Write down a similar rule to that in part b for the vector⎯→DE. Use this rule to work out

⎯→DE in

terms of a and b.

d Explain why C, D and E lie on the same straight line.

ABCDEF is a regular hexagon.⎯→AB is represented by the vector a,

and⎯→BC by the vector b.

a By means of a diagram, or otherwise, explain why⎯→CD = b – a.

b Express these vectors in terms of a and b.

i⎯→DE ii

⎯→EF iii

⎯→FA

c Work out the answer to⎯→AB +

⎯→BC +

⎯→CD +

⎯→DE +

⎯→EF +

⎯→FA

Explain your answer.

d Express these vectors in terms of a and b.

i⎯→AD ii

⎯→BE iii

⎯→CF iv

⎯→AE v

⎯→DF

A B

C

DE

F

a

b

B

O A

D

Ea

b

C

B

O A

C

a

b

568

CHAPTER 25: VECTORS



ABCDEFGH is a regular octagon.⎯→AB is represented by the

vector a, and⎯→BC by the vector b.

a By means of a diagram, or otherwise, explain why⎯→CD = √2b – a.

b By means of a diagram, or otherwise, explain why⎯→DE = b – √2a.

c Express the following vectors in terms of a and b.

i⎯→EF ii

⎯→FG iii

⎯→GH iv

⎯→HA

v⎯→HC vi

⎯→AD vii

⎯→BE viii

⎯→BF

In the quadrilateral OABC, M, N, P and Q are the midpoints

of the sides as shown.⎯→OA is represented by the vector a,

and⎯→OC by the vector c. The diagonal

⎯→OB is represented by

the vector b.

a Express these vectors in terms of a, b and c.

i⎯→AB ii

⎯→AP iii

⎯→OP

Give your answers as simply as possible.

b i Express the vector⎯→ON in terms of b and c.

ii Hence express the vector⎯→PN in terms of a and c.

c i Express the vector⎯→QM in terms of a and c.

ii What relationship is there between⎯→PN and

⎯→QM?

iii What sort of quadrilateral is PNMQ?

d Prove that⎯→AC = 2

⎯→QM.

L, M, N, P, Q, R are the midpoints of the line segments,

as shown.⎯→OA = a,

⎯→OB = b,

⎯→OC = c.

a Express these vectors in terms of a and c.

i⎯→OL ii

⎯→AC

iii⎯→OQ iv

⎯→LQ

b Express these vectors in terms of a and b.

i⎯→LM ii

⎯→QP

c Prove that the quadrilateral LMPQ is a parallelogram.

d Find two other sets of four points that formparallelograms.

A

N

O

CQ

L

M

PR

B

b

a

c

CN

B

P

AQO

M bc

a

A B

C

D

EF

G

H

a

b

569

CHAPTER 25: VECTORS



In triangle ABC, M is the mid-point of BC.

AB⎯→

= a and AC⎯→

= b

Find AM⎯→

in terms of a and b.

Give your answer in its simplest form.

The diagram shows two vectors a and b.

On a copy of the grid, draw the vector 2a – b

OAB is a triangle with X the mid-point of OA and Y themid-point of AB.

OA⎯→

= a and OB⎯→

= b

a Find, in terms of a and bi BA

⎯→

ii OY⎯→

b What type of quadrilateral is OXYB? Give a reason for your answer.

OPQ is a triangle. T is the point on PQ for which PT : TQ = 2 : 1

OP⎯→

= a and OQ⎯→

= b

a Write down, in terms of a and b, an expression

for PQ⎯→

.

b Express OT⎯→

in terms of a and b. Give your answerin its simplest form.

Edexcel, Question 19, Paper 5 Higher, November 2003

OPQ is a triangle.R is the midpoint of OP.S is the midpoint of PQ.

OP⎯→

= p and OQ⎯→

= q

a Find OS⎯→

in terms of p and q.

b Show that RS is parallel to OQ.

Edexcel, Question 21, Paper 5 Higher, November 2004

The diagram shows a regular hexagon ABCDEF with centre O.

OA⎯→

= 6aOB⎯→

= 6ba Express in terms

of a and/or b

i AB⎯→

,

ii EF⎯→

X is the midpoint of BC.

b Express EX⎯→

in terms of a and/or bY is the point on AB extended, such that AB : BY = 3 : 2

c Prove that E, X and Y lie on the same straight line.

Edexcel, Question 23, Paper 5 Higher, June 2003

O

A B

CF

E D

X

Diagram notaccurately drawn

6a 6b

q

p

R

S

O Q

P Diagram notaccurately drawn

T

Q

O Pa

b

Diagram notaccurately drawn

b

a

XY

A

BO

a b

A C

B

Ma

b

570



CHAPTER 25: VECTORS

571

WORKED EXAM QUESTION

Solution

a i⎯→BC =

⎯→BA +

⎯→AO +

⎯→OC = –2b – a + 6b = 4b – a

ii⎯→OQ =

⎯→OC +

⎯→CQ =

⎯→OC – 1–2

⎯→BC = 6b – 1–2 (4b – a) = 6b – 2b + 1–2 a = 1–2 a + 4b

iii⎯→PQ =

⎯→PB +

⎯→BQ =

⎯→PB + 1–2

⎯→BC = b + 1–2 (4b – a) = b + 2b – 1–2 a = 3b – 1–2 a

b⎯→PR =

⎯→PA +

⎯→AO +

⎯→OR = –b – a + 3b = 2b – a

So⎯→PS = 1–2

⎯→PR = b – 1–2 a

⎯→SQ =

⎯→SP +

⎯→PQ = 1–2 a – b + 3b – 1–2 a = 2b

⎯→OC = 6b, so

⎯→SQ is parallel to

⎯→OC

In the triangle OAB, P is the midpoint of AB, X is the

midpoint of OB, OA⎯→

= a and OB⎯→

= b. Q is the pointthat divides OP in the ratio 2 : 1.

a Express these vectors in terms of a and b.

i⎯→AB ii

⎯→AP

iii⎯→OP iv

⎯→OQ

v⎯→AQ vi

⎯→AX

b Deduce that⎯→AX = k

⎯→AQ, where k is a scalar, and

find the value of k.

A

O

BP

X

Qa

b

OABC is a trapezium with AB parallel to OC. P, Q and R are the mid-points of AB, BC and OC respectively. OC is three times the length of AB.⎯→OA = a and

⎯→AP = b

a Express, in terms of a and b, the following vectors.

i⎯→BC ii

⎯→OQ iii

⎯→PQ

b S is the mid-point of⎯→PR .

Prove that⎯→SQ is parallel to

⎯→OC.

A B

COR

P

Qa

b



572

GRADE YOURSELF

Able to solve problems using addition and subtraction of vectors

Able to solve more complex geometrical problems

What you should know now

� How to add and subtract vectors

� How to apply vector methods to solve geometrical problems



1 properties of this chapter will show you … vectors

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