ADDITIONAL MATHEMATICS FORM 5 MODULE 7 - … papers Add Maths/vector... · 2012-01-12 · 7.4 SPM...

VECTORS

ADDITIONAL MATHEMATICSFORM 5

MODULE 7

http://mathsmozac.blogspot.com

http://sahatmozac.blogspot.com

1

VECTORS

Contents Page

7.0 Conceptual Map 2

7.1 Introduction to VectorPractice 7.1

3

7.2 Multiplication of Vector By ScalarPractice 7.2.1Practice 7.2.2

4

5

7.3 Addition And SubtractionPractice 7.3

67-9

7.4 SPM Questions 10-12

Assessment test 13-14

Answers 15-16



2

7.0 CONCEPTUAL MAP

Parallel vectorsExample

(a) 2

AB + 3

AB = 5

(b) 4~a + 2

~a = 6

~a

(b) Parallelogram Law of vectorAdditions

PR =

PQ +

QR

=

PQ +

PS

VECTORSA vector is a quantity that hasa magnitude and a direction

Notation of Vectors

~aAB

~aBA

Magnitude │

AB │ or │-~a

(c) Polygon Law

AF =

AB +

BC +

CD +

DE +

EF

Expression of a vector as the linearcombination of a few vectors

DC =

DE +

EA +

AB +

BC

Multiplication of vector By a scalar

│ka│= k│~a │

AB =~a ;

PQ = 2

AB ate

= 2~a

Two vectors are parallel if one of the vectors isthe scalar multiple of the other vector

AB =2

1

PQ hence

AB //

PQ

A

B

A

B

P

Q Non-parallel vectors(a) Triangle Law of vectorAdditions

AC =

AB +

BC

AB

C

S

P Q

R

Addition and subtraction of vectors

F E

B

A D

Chttp://mathsmozac.blogspot.com


7.1 INTRODUCTIONS TO VECTOR

Find the magnitude and direction of

AB vector.

Distance of AB = 22 33

= 18

= 4.24 unit

Magnitude

AB = 4.24 unit

Direction = North-East

PRACTICE 7.1

Write the notation

Sharpen Your Skills 1

B


of vector and fin

Notation

AB5.0

B

AM

~a

A

http://m
3
d the magnitude and direction

Magnitude Direction99 unit North-East

F

G

H

E

D

K

C

Q

P

~b

~c

~d

~

f~e

athsmozac.blogspot.com

7.2 MULTIPLICATION OF VECTOR BY SCALAR

1. Express

AB and

PQ as a scalar product of~a

Solution

AB = 2 a


PB


4

~

PQ =2

11

~a

PRACTICE 7.2.1

1. In the figure, ACEG is a parallelogram. It is given that AH

=~a ,

AB =~b and

AJ

=~c . Points B, D, F, H are the midpoints of AC, CE, EG and AG

respectively. Find each of the following vectors in terms of~a ,

~b and

~c

(a) AE

=

(b) AC

=

(c) AG

=

(d) FB

=

(e) HD

=

Given AB

=~u and B C

=~

4u , shows that A, B and C are collinear

AB

=~u , B C

=~

4u

B C

= 4 AB

AB

B C

Point B is common point. A, B and C are collinear

~c

G

JH

A B C

D

F E

~b

~a


~a

Q

A


PRACTICE 7.2.2

1.Find the parallel vector of each of thefollowing

2.

3.

R

S~

4 w

~3u

~2u


Figure below shows a t

that DC

=~

6u , AB

=

Given that AB

AB

= 4

collinear.

QP~

3v

D

A

http:
5
rapezium ABCD in which AB

and DC

are parallel. Given

4 cm and DC

= 3 cm, express AB

in terms of~u .

~x and BC

= 6~x . Shows that the point A, B and C are

B

C

~3v

~6u

//mathsmozac.blogspot.com

6

4. Given that (2h – 3)~a = (k + 5)

~b , find the value of h and k if

~a and

~b are non-

parallel and non-zero vectors.

7.3 ADDITIONS AND SUBTRATIONS VECTORS

1.~a + 2

~a +2

~a Find the sum of the vectors

Sum of vectors = 10~a

2.

=~ ~

x y

3.

A C A B B C

=~ ~

( )a b

=~ ~a b

Q

P R

Q

~x 2y

B

~a

~b

A C

B

~a

~b

A C


P R

P R P Q Q R



7

PRACTICE 7.3

1.

In the above figure, point D is on BC,(a) Write

(i) DA

in term of CA

and DC

(ii) A D

in term of BD

and AB

(b) Find the resultant vector for CA

+ BC

2.

In the figure, ABCD is a parallelogram which diagonals AC and BD intersecteach other at point E. Find the resultant vector for each of the following.

(a) AB BD

(c) CA BC

(b) CE ED

(d) EB DE

BA

D

C

CD

A B

E



8

3.

Based on the above figure, BCDE is a straight line with BC CD DE

. Given

that~ ~

3 2AB x y

and~ ~

BC x y

. Express each of the following vectors in terms

of~ ~

/a and or b

(a) ED

(a) AC

(a) DA

4.

In the above figure, ABCDEF is a regular hexogen. Find the resultant vector foreach of the following

(a) AB BC CD

(b) AC DE CD

(c) AB CD EF BC DE FA

A

C

B

D

E

~ ~

x y

~ ~

3 2x y

C D

B E

FA



9

5.

In the above figure,~

BA a

and~

BC b

. If Q is a point above BC

so that

: 1:3BQ QC

, express QA

in terms of~ ~a and b .

A

B C

~a

~b



10

7.4 SPM QUESTION

1. SPM 2003

P = 2a + 3bq = 4a – br= ha + (h – k)b, where h and k are constants

Use the above information to find the values of h and k when r = 3p – 2q[3 marks]

2. Diagram shows a parallelogram ABCD with BED as a straight line.

Given that~ ~

6 , 4 and DE = 2EB,AB p AD p

express in terms of p and q

(a) BD

(b) EC

[ 4 marks]

BA

E

CD



11

3. SPM 2004

Diagram below shows triangles OAB. The straight line AP intersects the

straight line OQ at R. It is given that1

3OP OB ,

~ ~

1, 6 2

4AQ AB OP x and OP y

(a) Express in terms of~x and/or

~

:y

(i) AP

,

(ii) OQ

[4 marks]

(b) (i) Given that AR h AP

, state AR

in terms of h,~x and

~

y

(ii) Given that RQ k OQ

, state RQ

in terms of k,~x and

~

y

[2 marks]

(c) Using AR

and RQ

from (b), find the value of h and of k

[4 marks]

R

BP

Q

O

A



12

4. SPM 2005

In diagram, ABCD is a quadrilateral. AED and EFC are straight lines.

given that~

20AB x

,It is

~ ~~

18 , 25 24 ,

4AE Y DC x y AE AD

and3

5EF EC

(a) Express in terms of~x and/or

~

:y

(i) BD

,

(ii) EC

[3 marks](b) Show that the points B, F and D are collinear [2 marks]

(c) If~ ~

2 3x and y , find BD

[2 marks]

A

CEF

B

D

•



13

ASSESSMENT

1. In the above figure, GH : AB = 3 : 10 and GH is parallel to AB

. If AB= 10~a ,

find GH

in terms of~a . [ 2 Marks]

2.

In the above figure PQRSTU is a regular hexagon. Express PQ

+ PT

- RS

as a

single vector.

A

HG

C

B

U

P

ST

R

Q



14

3. Given AB

= (k + 1)~a and BC

= 2~b . If A, B and C are collinear, AB

= BC

and

~b = 3

~a . Find the value of k. [ 3 Marks]

4. Given that OABC is a rectangle where OA = 6 cm and OC = 5cm. If OA

=~a

andOB

=~b , find [ 3 Marks]

(a) AC in terms of~a and

~b

(b)~

~

a b

5. In OPQ, OP

=~

p and OQ

=~

q . T is a point on PQ where PT : TQ=2 : 1. Given

that M is the midpoint of OT, express PM

in terms of~

p and~

q . [ 3 Marks]



15

ANSWERS

PRACTICE 7.1

notation magnitude direction

CD 2.828 North west

HG 3 North

FE 3.162 South-west

MK 2 East

QP 2.236 South-west

PRACTICE 7.2.1 PRACTICE 7.2.2

1. (a) 2~c 1. PS

QR

(b) 2~b 2. 8

~u

(c ) 2~a 3. h =

3

2, k= -5

(d) -2~a

(e) 2~b

PRACTICE 7.3

1. (a) (i) DC

+ CA

4. (a) AD

(ii) AB

+ DC

(b) AE

(b) BA

(c )~0

2. (a) AD

5.~

1

4b +

~a

(b) CD

(c) BA

(d) DB

3. (a)~

y~x (c)

~5 x

(b) 4~x +

~

y



16

PAST YEARS QUESTION

1. SPM 2004

(a) (i) AP

=~

2 y + 6~x

(ii) OQ

=~

3

2y +

~

9

2x

(b) (i) AR

= h( 6~x - 2

~

y )

(ii) RQ

= k(~

9

2x +

~

3

2y )

(c) k =1

3, h =

1

2

2. h = -2 , h = -133. (a)

~ ~

6 4p q

(b)~ ~

82

3p q

2. SPM 2005

(a) (i) BD

=~

20 x + 32~

y

(ii) EC

= 25~x

(c) AD

= 104

ASSESSMENT

1. 3~a

2. PR

3. k = 54. (a)

~b - 2

~a

(b) 13 units

5.~

1

3q +

~

5

6p



17

VECTORS

ADDITIONAL MATHEMATICSFORM 5

MODULE 8



18

VECTORS

Contents Page

8.0 Conceptual Map 2

8.1 Vector In The Cartesian Coordinates 3 - 7

8.2 SPM Questions 8 – 10

8.3 Assessment test 11 - 12

Answers 13 - 14



19

8.0 CONCEPTUAL MAP

~

j

~i

~r

~

y j

~x i 1,1

x

y

VECTOR

The Magnitude

2 2

~r x y

The Unit Vector

~

2 2 ~ ~~~

1( )

rx i y j

x yrr

For any two points P and Q,the position vector of Qrelative to P is defined

by PQ OQ OP

, where O

is the origin.

VECTOR IN THE CARTESIANCOORDINATE

~i =

1

0

,~

0

1j

A vector~ ~ ~

xr x i y j

y



8.1 VECTOR IN THE CARTESIAN COORDINATES

1. State the following vector in terms in~i and

~

j and also in Cartesian coordinates

Example Solutions

22OA i

5


~ 0

~

03

3OB j

~ ~

3 4

3

4

OP p i j

Exercise Solutions

(a) OP

= (b) OQ

(c) OR

(d) OS

~

j4

3

2

1

543210

~

p

~i

B

P

A

1

4

3

2

1

2

SR

P

Q

-1-3 -2 -1

T

W

31 4

~

j

-2

O

http://
20
(e) OT

(f) OW

~i

mathsmozac.blogspot.com

21

2. Find the magnitude for each of the vectors

Example

3~ ~

2i j =2 23 2

13 unit

(a)~ ~

2 5i j

(b)~ ~

5 12i j (c)~i j =

3. Find the magnitude and unit vector for each of the following

Example

~ ~ ~

3 4r i j

Solution :2 2

~

~ ~ ~

Magnitude, 3 4

= 5

1unit vector, r, (4 3 )

5

r

i j

(a)~ ~ ~

2 6r i j

(b)~

6

3a

(c)~

1

2h



22

4. Given that~ ~ ~ ~~ ~

3 2 , b 2 5a i j i j and~ ~ ~

4 3 . Simplifyc i j of the following

(a)~ ~ ~a b c (b)

~ ~ ~2 3a b c

5. Given that~ ~ ~ ~ ~ ~~ ~ ~

2 3 , b 4 and c 2 5a i j i j i j . Determine the unit vector in the

same direction with vector~ ~ ~

2 3a b c .

6. If~ ~~ ~ ~

3 , OB 2 and BC 4 3OA i j j i j

. Find

(a) OC

(b) AC



23

ExampleGiven that

~ ~ ~ ~~ ~

6 7 and b 2a i j p i j .find the

possible value (or values) of p for each ofthe following cases.(a)

~ ~and ba are parallel.

(b)~ ~a b

Solutions

~ ~

~ ~~ ~

~ ~~ ~

6 7 ( 2 )

6 7 2

a k b

i j k p i j

i j kp i k j

By equating the coefficients of~i

6 = kp ………(1)

By equating the coefficients of~

j

7 = 2k ……….(2)

k =7 1

32 2

Subtitute k =7

2into (1)

6 =7

2p

p =12 5

17 7

(b)~ ~a b

2 2 2 2

2

6 7 2

85 4

p

p

p² +4 = 85p² = 81p = 9

7. Given~ ~ ~

4 ( 3)r i k j and

~ ~ ~

( 4) 14s k i j . If~r is parallel to

~s ,

find(a) the value of k

(b)~~

3s r for the positive values of k



24

ExampleO is the origin, P and Q are two points such

that2

3OP

and

2 3

-6

k kOQ

.

If the points O, P and Q are collinear, findthe possible values of k.

SolutionsSince the points O, P and Q are collinear,

OP mOQ

( m is a constant)22 3

3 -6

k km

22 ( 3 )

3 -6m

m k k

m(k² - 3k) =2 ……..(1)-6m = -3 …………...(2)

m =1

2

From (1):1

2(k² - 3k) =2

k² - 3k = 4k² - 3k- 4 = 0

(k – 4 )( k + 1 ) = 0k = 4 or -1

8. Given that~ ~

6 8PQ i j

and

~ ~

2PR i b j

.If P, Q and R are collinear,

find the value of b.



25

8.2 SPM QUESTIONS

SPM 2003/no. 12 / paper 1.

1. Diagram 2 shows two vectors, and QOOP

.

Express

(a) OP

in the formx

y

,

(b) OQ

in the form xi + yj. [ 2 marks]

SPM 2005 / no. 15 / Paper 12. Diagram 5 shows a parallelogram, OPQR, drawn on a Cartesian plane.

It is given that OP

= 6i + 4j and PQ -4i 5j

. Find PR

. [3 marks]

●P(5, 3)

y

●Q(-8, 4)

xO

R

Q

P

O x

y



26

SPM 20043. Given that O(0, 0), A( -3, 4) and B( 2, 16), find in terms of the unit vectors,

~ ~

and ji

(a) AB

(b) the unit vector in the direction of AB

[ 4 marks]

SPM 20044. Given that A( -2, 6), B( 4, 2) and C( m, P), find the value of m and of p such that

~ ~

2 10 12AB BC i j

. [ 4 marks]



27

SPM 2003

5. Give that5 2

, OB and CD7 3 5

kAB

, find

(a) the coordinates of A [ 2 marks]

(b) the unit vector in the direction of OA

[ 2 marka]

(c) the value of k, if CD

is parallel to AB

[ 2 marks]



28

8.3 ASSESSMENT TEST

1. Find the magnitude and unit vector of~~ ~

= 6 i 8a j [2 marks]

2. Find the unit vector in the direction of12

5OP

[2 marks]

3. Given that~ ~ ~

3u i m j and~ ~ ~

3v k i j , find the value of m and of k if

~ ~ ~ ~

11 4u mv i j [3 marks]

4. Three points A, B and C are such that5 6

, and OC6 4 2

kOA OB

, where O

is the origin. Find the value of k if the points A, B and C are collinear [4 marks]



29

5. It is given that~ ~ ~ ~~ ~

2 2 , 2 ,a i j b i j P(1, -3) and Q(5, 2). If~ ~

PQ h a k b

, where

h and k are constants, find(a) the value of h and k [4 marks]

(b) the unit vector in the direction of PQ

. [2 marks]

6. PQRS is a parallelogram. M is the midpoint of QR. Given that~ ~

3PQ i j

and

~ ~

4 2PM i j

, find

(a) (i) QR

(ii) PR

[4 marks](b) the length of PS [2 marks]

7 .Given that~ ~

7 11and v

9 3u

k

, find the possible value(s) of k for each of the

following cases.(a)

~ ~and vu are parallel [3 marks]

(b)~ ~u v [3 marks]



30

ANSWERS

Exercises Assesment

1. (a)~ ~

3OP i j

1.~ ~

3 4

5 5i j

(b)~ ~

3 2OQ i j

2.

12

13

5

13

(c)~ ~

2OR i j

3 m =2, k = 4

( d)~ ~

2 2OS i j

4 k = 7

(e)~

4OT i

5 (a) k =3, h =1

2

(f)~

2OW j

(b)~ ~

1(4 5 )

41i j

2. (a) 13 units 6 (a) (i)~ ~

2 2i j

(b) 13 units (ii)~ ~

5 3i j

( c) 2 units (b) 2.828 units

3. (a) 40 units ,~ ~

1(2 6 )

40i j 7 k =

117

7

(b) 45 units,61

345

k = 0 or 6

(c) 5 units,11

25

4. (a)~ ~

9 6i j

(b)~ ~

19 17i j

5.~ ~

1( 5 23 )

554i j

6. (a)~ ~

4 5i j

(b) 89



31

7. (a) k = -11, 4

(b) 6 units

8. b =8

3

SPM QUESTIONS

1. (a)5

3OP

(b)~ ~

8 4OQ i j

2.~ ~

10 i j

3. (a)5

2

(b)51

1213

4. m = 6, p = -2

5. (a) A 3,4

(b)31

45

(c) k =25

7



ADDITIONAL MATHEMATICS FORM 5 MODULE 7 - … papers Add Maths/vector... · 2012-01-12 · 7.4 SPM...

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