FormFour Mather Notes

8/16/2019 FormFour Mather Notes

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Example 2

"ind the solution to the following system of simultaneous equations by graphical

method.


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Simultaneous Equations GraphicallySolve simultaneous equations graphically


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Example

#olve the following simultaneous equations graphically and check your solution by a

non-graphical method


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Example !

"ind the solution to the following system of simultaneous equations by graphical

method.


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Exe"#ise 1


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"ind the solution to the following systems of simultaneous equations graphically.

Try $li paid %& shillings for '( oranges and %) mangoes. *oshi went to the same

market and paid +& shillings for ', oranges and ' mangoes. hat was the price for a

mango and for an orange/

Linea" inequalities

• !ormally any straight line drawn on xy – plane separates it into twodis"oint sets. These sets are called half – planes

• #onsider the equation y $ % drawn on the xy plane as shown below.

"rom the figure above0 all points above the line0 that is all points in the half plane $which is above the line satisfy the relation y1) and those lying in the half plane 2which is below the given line0 satisfy the relation y3 ).


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S$a%in& o' Re&ions

• In linear programming usually the region of interest is left clear that iswe shade unwanted region&s'.

NB:

hen shading the half planes we consider the inequalities as the equations but dottedlines are used for the relations with 1 or 3 signs and normal lines are used for thosewith 4 or 5 signs.

6onsider the inequalities x1(0 y1( and +x 7 %y 1'+ represented on the xy-plane 8n thiscase we draw the line x9(0 y9 ( and +x7%y9'+ but the point about the inequality signsfor each equation must be considered.


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Feasi)le Re&ion

Definition: 8n the xy plane the region that satisfies all the given inequalities is calledthe feasible region (F.R)

Example *

8ndicate the feasible region for the inequalities +x7%y 4 '+ and y-x 5 +.


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:etermine the solution set of the simultaneous inequalities y 7 x 4% and x-+y 5 ;.


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Example +

"atuma was given %( shillings to buy oranges and mangoes. $n orange costs +shillings

while a mango costs % shillings. 8f the number of oranges bought is at least twice the

number of mangoes0 show graphically the feasible region representing the number of

ranges and mangoes she bought0 assuming that no fraction of oranges and mangoes

are sold at the market.Solution:-

Le x be the number of oranges she bought and y the number of mangoes she bought.

i? and x4+y =====.. >ii?0

$lso because there is no negative oranges or mangoes that can be bought0

then x4 and y4( ===.. >iii?


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(0 '(? and >')0(? and the linex4+y or x @ +y 4 ( is the line which passes through >(0(? and >+0'?.

Exe"#ise 2

"or practice.

(. Draw the graph of the equation )x – y $ * and show which half plane is

represented by )x – y +* and the one represented by )x – y *

). -n the same coordinate axes draw the graphs of the followinginequalities x )y / ), y0x / ( and y 1 2.

3. Draw the graphs of y )x 0( and y + 3 – x on the same axes andindicate the feasible region.

4. 5 post o6ce has to transport 7*2 parcels using a lorry, which ta8es(%2 parcels at a time and a van which can ta8e 92 at a time. The cost ofeach "ourney is 3%2 shillings by lorry and )72 shillings by van. The vanma8es more trips than the lorry and the total cost should not exceed 3272

shillings. :how graphically the feasible region representing the number oftrips that a lorry and a van can ma8e.

The Objective unction!n Objective unction from Word Problems

Form an objective function from word problemsLinear programming components


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$ny linear programming problem has the following

a. -b"ective

b. 5lternative course &s' of action which will achieve the ob"ective.

c. The available resources which are in limited supply.

d. The ob"ective and its limitations should be able to be expressed aseither linear mathematical equations or linear inequalities. Therefore linearprogramming aims at ;nding the best use of the available resources.

Arogrammingis the use of mathematical techniques in order to get the best possiblesolution to the problem

Steps to )e 'ollo,e% in sol-in& linea" p"o&"ammin& p"o)lems

a.


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Therefore the student will buy , exercise books from each site.

Example 0

$ nutritionist prescribes a special diet for patients containing the following number of

Cnits of vitamins $ and 2 per kg0 of two types of food f ' and f +


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8f the daily minimum in take required is '+( Cnits of $ and D( units of 20 what is theleast total mass of food a patient must have so as to have enough of these vitamins/

Solution

Let x be the number of kg>s? of "' that patient gets daily and y be the number of kg>s?of "+ to be taken by the patient daily.

ObEective function " >x0 y? 9 >x 7 y? minimum

f >6? 9 '( 7 ( 9 '(

#o f >2? 9 ,. is the minimum


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Therefore the least total mass of food the patient must have is 6. !ilograms

The %inimum and %a&imum 'alues usin$ the Objective unctioFind the minimum and ma"imum values using the objective function

Example 1

$ farmer wants to plant coffee and potatoes. 6offee needs % men per hectare whilepotatoes need also % men per hectare. Be has & hired laborers available. To maintain

a hectare of coffee he needs +)( shillings while a hectare of potatoes costs him '((

shillings. ."ind the greatest possible land he can sow if he is prepared to use +)0((( shillings.

Solution

Let x be the number of hectares of coffee to be planted and y be the number ofhectares of potatoes to be planted.

ObEective function f >x0 y? 9 >x0 7 y? maximum

%x 7 %y 5 & or x 7 y 5', ====.>i?

+)(x 7 '((y5 +)0((( Or )x 7 +y 5 )((===>ii?

x 4 ( =======...>iii?

y4 ( =======...>iv?

Csing the obEective function f >x0 y? 9 >x 7 y? maximum0


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f >$? 9 >( 7 +)(? 9 +)(

f >2? 9 >(7',? 9 ',

f >6? 9 >',7(? 9 ', 3F--Gnd"ragment--1

f>:? 9 >'((7(?9 '(( >maximum?

Therefore the greatest possible area to be planted is +)( hectors of potatoes.

N3: 8n most cases L.A problems must involve non-negativity constraints >inequalities?that are x 4 ( and y 4 (. This is due to the fact that in daily practice there is no use of negative quantities.

Example 11

$ technical school is planning to buy two types of machines. $ lather machine needs

%m+ of floor space and a drill machine needs +m+ of floor space. The total space

available is %(m+

. The cost of one lather machine is +)0((( shillings and that of drillmachine is %(0((( shillings. The school can spend not more than %((0((( shillings0

what is the greatest number of machines the school can buy/Solution:

Let x be the number of Lather machines and y be the number of drill machines to bebought

ObEective function: f>x0 y? 9 >x 7 y? max

8nequalities:

%x 7 +y 5 %(.. =======.>i?

+)0(((x 7 %(0(((y 5%((0(((

Or )x 7 ,y 5 ,(========..>ii?

x 4 ( ============.>iii?

y 4 (======== ====.>iv?


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#ince the incomplete machine canHt work0 then 2 9 >0 %? or >D0 &?.That is

approximating values of x and y to the possible integers without affecting the giveninequalities or conditions.

$? 9 ( 7 '( 9 '(

f>2? 9 D 7 & (r f >2? 9 7 % 9 ''

f >6? 9 '( 7 ( 9 '(

f >:? 9 ( 7 ? 9 (

#o f >2? gives the maximum number of machines which is ''.

Therefore the greatest number of machines that can be bought by the school is ''machines.

Exe"#ise

'. #how on a graph the feasible region for which the restrictions are


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y 5 +x0 x4 ,0 y4+ and +x 7 %y 5%(

"rom the graph at which point does

a. y – x ta8e a maximum valueA

b. x y ta8e a maximum valueAc. y – x ta8e a maximum valueA

+. ith only +(0((( shillings to spend on fish0 Iohn had the choice of buying two types

of fish. The price of a single fish type ' was +0)((shillings and each fish of type + was

sold at +0((( shillings. Be wanted to buy at least four of type '. hat is the greatest

number of fish did Iohn buy/ Bow many of each type could he buy/

%. Bow many corner points does the feasible region restricted by the inequalities/

x4(0 y 4 (0 %x 7 +y 5 ' and +x 7 &y 5', have/

hich corner point maximizes the obEective function f >x0 y? 9 +x 7 )y/

P#O(!(ILIT)s?

Definition Arobability set is the set of all outcomesJresults from the experimentbeing performed.

"or example when tossing once a fair coin the expected outcomes are either head>B?or tail>T? to be shown up.

8n this case the probability set is


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# 9 KB0 T

$lso if a fair die is tossed once what is expected to show up is only one number amongthe six numbers0 that is '0+0%0&0)0,.


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Solution

#9 K'0 +0 %0 &0 )0 ,0 D0

G 9 K+0 &0 ,0

#o GN 9 K'0 %0 )0 D

where G is the event of selecting an even number and GN is the event of not selectingeven number less than ;.

Exe"#ise 1

'. rite theprobability set of each of the following experiments

a. 5 die is tossed and the face showing up is read.

b. 5 friend is as8ed for the month of his birth.

c. The sex of a human being is as8ed.d. 5 card is drawn from a box containing ;ve cards bearing the numerals

),4,9,7 and (2.

+. rite inset notation the elements of the following events

a. 5 fair die is rolled and the number obtained is greater or equal to %.

b. 5 prime number between )2 and 42 is chosen.

%. rite inset notation the elements of the event of not choosing an even number

between +) and ))

E&perimental #esults in #elation to #eal Life Occurrences%nterpret e"perimental results in relation to real life occurrences

"or example when tossing once a fair coin the expected outcomes are either head>B?or tail>T? to be shown up.

8n this case the probability set is

# 9 KB0 T

$lso if a fair die is tossed once what is expected to show up is only one number amongthe six numbers0 that is '0+0%0&0)0,.


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Definition: The probability of an event is the ratio between the number of times theevent has occurred to the total number of experiments that have been done.

8f A>G? 8s the probability of the event G0 then

$lso the probability found by experimenting is referred to as experimental probability.

The ormula to "alculate the Probability of an Event 'pply the formula to calculate the probability of an event

Example !

$ drawing pin was tossed '((( times. The number of tosses where the pin fell flat was

),%. 6alculate the probability that when such a pin is tossed0 it will fall flat.

Example (

) of torch bulbs manufactured by a certain factory were defective. hat is the

probability that when a bulb from that factory is tested it will be defective/Solution

A>G? 9 ) 9 )J'(( 9 (.()

unbiased?


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Example *

$ piece of chalk is picked from a box containing ) identical pieces two of which are

red and the remaining are white. "ind the probability that the piece of chalk picked is

red

Example +

"ind the probability that a ling appears in a drawing a single card from an ordinary

deck of )+ cards.


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Example /

hat is the probability of not getting an even number when a fair die is tossed/


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Example 0

hat is the probability of selecting a green ball from the box containing red and

green balls if the probability of selecting red ball is 'J&/

Example 1

hen tossing a die what is the probability of getting a number greater or equal to '/


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8f two or more simple events may occur or take place at the same time then theevents are combined events.

"or instance when the experiment of tossing two coins at the same time is done0 thenthe event of interest canHt simply be determined.

Let G 9 KObtaining two heads

8n this case there are two simple events which are obtaining the head on the first coinand obtaining the head on the second coin.

#o G' 9 KObtaining the head on the first coin

G+ 9 KObtaining the head on the second coin

! Tree +ia$ram of "ombined Events#raw a tree diagram of combined events

The event G can be found by using what is referred to as a tree diagram.

Therefore0 # 9 K>B0 B?0 >B0 T?0 >T0 B?0 >T0 T? is the sample space.

Example 11

$ die and 6oin are tossed together. :raw a tree diagram to find the #ample space and

hence determine the probability that a head and a number less than % occurs.


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Example 12

$ fraction is written by selecting the numerator from the digits '0 +0 % and the

denominator from the digits ,0 .

a. Draw a tree diagram to ;nd the sample space of this experiment.

b. ?ind the probability that a the fraction written is less than K


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Example 1

8n a family of % children what is the probability that

a. 5ll are girls

b. 5t least two are boys


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Example 1!

Three coins are tossed simultaneously. "ind the probability that

a. 3 heads appear

b. ) tails and one head appear


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Exe"#ise

'. 8f two digitsnumeral is written choosing tenHs digits from the set K'0 +0 %0 &0 and

the unitHs digit from K)0, what is the probability that a number greater than +( willappear/

+. $ pair of dice istossed. "ind the probability that the sum of the two numbers

obtained is

a. 5t least 7

b. at most (

c. xactly 9

%. 8n a familywith two children0 what is the probability that

a. Loth are boys

b. 5t least one is a boy

&. $ die and twocoins are tossed at the same time find the probability that

a. 5 number 9 and two heads will appear.

b. 5 number less than 4, a head and tail will appear.

c. 5 number multiple of ) and two tails will appear.

The Probability of T*o "ombined Events usin$ the ormulaFind the probability of two combined events using the formula

Mutuall6 Ex#lusi-e E-ents

Two or more events are said to be mutually exclusive if the occurrence of one eventhinders the occurrence of the other. This means that for mutually exclusive events0only one event may occur at a time0 e.g.0 it is impossible for two numbers say ' and ,on a single die to show up for one tossing.


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Therefore if $ and 2 are two events0 then the probability of $ or 2 is given by

Example 1(

8f in a class there are %& students instead of %) and 8ssa0 anna0 Gliza and Iuma apply

for the one chance remaining what is the probability that either $nna or Iuma will be

chosen/

Example 1*

"ind the probability that an even or an odd number which is greater than ' occurs

when a die is tossed once.

Example 1+

The following table shows years of experience for plumbers in a builders company.


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Independent Eents

Definition: Two or events are said to be independent events if the occurance of oneevent does not affect the occurrence of other event>s?

"or example when a die and a coin are tossed together0 the occurrence of a tail onthe coin does not hinder the occurrence of the number ) on the die.

Example 1/

$ die and coin are tossed. "ind the probability that a number greater than & appears

on the die and a tail appears on the coin


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Example 10

$ box contains ; oranges0 D mangoes and + lemons. $ fruit is drawn from the box and

then replaced. $nother draw is made. hat is the probability that both fruits drawn

are mangoes.

Example 2

The probability that a man and his wife will be alive for )( years are %J'( and 'J%

respectively. "ind the probability that


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Exe"#ise !

#elf test.

(. 5 coin is tossed and a card is drawn from an ordinary pac8 of%)cards.?ind the probability that an ace is drawn and a head is obtained onthe coin &There 4 aces in a pac8 of cards'

). Two numbers are selected from the integers ( to (( inclusively,repeation being allowed. ?ind the probability that &a' Loth prime &b' Loth arepowers of )

%. 8n the village0the probability that a man selected at random on a #unday morning iscarrying more than is (.D. "ind the probability thatM

a. Two men selected at random on a :unday morning is carrying morethan 328g

b. Three men selected at random are all carrying more than 328g

&. $ letter is chosenfrom the word PrandomQ hat is the probability that it is an n or

d/

). >a? hat does itmean by saying that the probability of an event is >i? ( >ii? ' >b?

!ive two examples of impossible of events.


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%!T#I"ES !,+ T#!,SO#%!TIO,S

s farmed0 this can be done by removing the headings and the bracket enclosing thenumbers >elements? and given a name >normally a capital letter?.

%? rows and four >&? columns.

8n the matrix $0 %& is the element >entity? in the second row and third column while+ lies in the first row and fourth column. The plural form of matrix is matrices.


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Types of "atri%es:

The following are the common types of matrices-


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%atrices of order up to - . - 'dd matrices of order up to ) * )

hen adding or subtracting one matrix from another0 the corresponding elements>entities? are Jadded or subtracted respectively.

This being the case0 we can only perform addition and subtraction of matrices withthe same orders.

Example 1

!iven that

%atrices of order up to - . -Subtract matrices of order up to ) * )

Example 2

!iven that


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Example

#olve for x0 y and z in the following matrix equationM

Exe"#ise 1

:etermine the order of each of the following matricesM

+. !iven that

%. !iven that


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&. $ house wife makes the following purchases during one week *onday +kg of meat

and loaf of bread ednesday0 'kg of meat and #aturday0 'kg of meat and one loaf of

bread. The prices are ,(((J9 per kg of meat and )((J9 per loaf of bread on each

purchasing day

a. Hrite a 3x) matrix of the quantities of items purchased over the threedays .

b. Hrite a )x( column matrix of the unit prices of meat and bread.

). #olve for x0 y and z in the equation

$dditive identity matrix.

8f * is any square matrix0 that is a matrix with order mxm or nxn and R is another

matrix with the same order as m such that

*7 R9 R7* 9 * then R is the additive identity matrix.

The additive inverse of a matrix.

8f $ and 2 are any matrices with the same order such that $72 9 R0 then it means thateither $ is an additive inverse of 2 or 2 is an additive inverse of $ that is 29-$ or $9 -2


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Example !

"ind the additive inverse of $0

Example (

"ind the additive identity of 2 if 2 is a %S% matrix.

! %atri& of Order - . - by a Scalar+ultiply a matri" of order ) * ) by a scalar

$ matrix can be multiplied by a constant number >scalar? or by another matrix.

S%alar "ultipli%ation of "atri%es:

Rule: 8f $ is a matrix with elements say a0 b0 c and d0 or

Example *

!iven that

#olutionM


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Example +

!iven0

Solution$


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T*o %atrices of order up to - . -+ultiply two matrices of order up to ) * )

+ultiplication of +atri" by another matri":

$2 is the product of matrices $ and 2 while 2$ is the product of matrix 2 and $.

8n $20 matrix $ is called a pre-multiplier because it comes first while matrix 2 iscalled the post multiplier because it comes after matrix $.

Rules o' 'in%in& t$e p"o%u#t o' mat"i#es

(. The pre –multiplier matrix is divided row wise, that is it is dividedaccording to its rows.

). The post multiplier is divided according to its columns.

3. @ultiplication is done by ta8ing an element from the row and multipliedby an element from the column.

4. In rule &iii' above, the left most element of the row is multiplied by thetop most element of the column and the right most element from the row is


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multiplied by the bottom most element of the column and their sums areta8en

Therefore it can be concluded that matrix by matrix multiplication is only possible ifthe number of columns in the pre-multiplier is equal to the number of rows in thepost multiplier.

Example /

!iven ThatM


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"rom the above example it can be noted that $22$0 therefore matrix by matrixmultiplication does not obey commutative property except when the multiplicationinvolves and identity matrix i.e. $898$9$

Example 0

Let0


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Example 1

"ind 6S: if

(roduct of a matri" and an identity matri":


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%.Csing the matrices

&."ind the values of x and y if

Inverse of a %atri&The +eterminant of a - . - %atri&

,alculate the determinant of a ) * ) matri" :eterminant of a matrix


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Example 12

"ind

Example 1

6onsidering

Example 1!

"ind the value of x


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Singular and non singular "atri%es:

:efinition

$singular matrix is a matrix whose determinant is zero0 while non @ singular matrix isthe one with a non zero determinant.

Example 1(

"ind the value of y


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The Inverse of a - . - %atri&Find the inverse of a ) * ) matri"

In-e"se o' mat"i#es

7e'inition 8f $ is a square matrix and 2 is another matrix with the same order as $0then 2 is the inverse of $ if $292$98 where 8 is the identity matrix.

Thus $292$98 means either $ is the inverse of 2 or 2 is the inverse of $.

here 29$-'0 that is 2 is the inverse of matrix $


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#ince we need the unknown matrix 20 we can solve for p and q by using equations >i?and >iii? and we solve for r and s using equations >ii? and >iv?

To get p proceed as follows


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$lsoto get r and s0 the same procedure must be followed

$nd


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Example 1+

hich of the following matrices have inverses/


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Exe"#ise

'. "ind the determinant of each of the following matrices.

+. hich of the following matrices are singular matrices/

%. "indinverse of each of the following matrices.


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- . - %atri& to Solve Simultaneous Equations 'pply ) * ) matri" to solve simultaneous equations

#olving simultaneous equations by matrix method


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Then 29 $-'S6

Example 1/

2y matrix method solve the following simultaneous equations


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*ultiplying $-' an each side of the equation0 gives0


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Example 10

#olve


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*ultiplying $-' on each side of the equation gives0


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Example 2

2y using matrix method solve the following simultaneous equations

*ultiplying $-' on each side of the equation gives0


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6ramerHs Xule

#o


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Example 21"ind

Example 22

2y using 6ramerHs rule


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Example 2

2yusing 6ramerHs rule0

Exe"#ise !'. Cse the matrix method to solve the following systems of simultaneous equations.


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Cse 6ramerHs rule to solve the following simultaneous equation

%. hythe system of simultaneous equations

%atrices and Transformations:efinition $ transformation in a plane is a mapping which moves an obEect from oneposition to another within the plane. "igures on the plane can also be shifted fromone position by a transformation.

$ new position after a transformation on is called the i"age.

Gxamples of transformations are >i? Xeflection >ii? Xotation >iii? Gnlargement >iv?Translation.

!ny Point P/.0 )1 into P2/.20)21 by Pre3%ultiplyin$ / 1 *ith a TransformationMN%atri& T

Transform any point (-* /0 into (1-*1/10 by pre2multiplying - 0 with a ᵡᵧtransformation matri" T

- #uppose a point A>x0y? in the x-y plane moves to a point AY >xY0yY? by atransformation T0


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$ transformation in which the size of the image is equal that of the obEect is called an

8#O*GTX86 *$AA8

The %atri& to #eflect a Point P/.0 ) 1 in the .3!&is 'pply the matri" to reflect a point (-* / 0 in the "2a"is

Re'le#tionM

hen you look at yourself in a mirror you seem to see your body behind the mirror.Zour body is in front of the mirror as your image is behind it.

$n obEect is reflected in the mirror to form an image which isM

a. The same size as the ob"ect

b. The same distance from the mirror as the ob"ect

#o reflection is an example of 8#O*GTX86 *$AA8


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The mirror is the line of symmetry between the obEect and the image.

Example 2!

"ind the image of the point $ >+0%? after reflection in the x @ axes.#olutionM

Alot point $ and its image $Y such that $$Y crosses the x @ axis at 2 and alsoperpendicular to it.

"or reflection $2 should be the same as 2$Y i.e. $2 9 2$Y

"rom the figure0 the coordinates of $ Y are $Y >+0-%?. #o the image of $ >+0%? underreflection in the x-axis is $Y >+0-%?

+0%? 9- >+0-%?.

here *x means reflection in the x @ axis and *y means reflection in the y-axis.

The %atri& to #eflect a Point P/.0 )1 in the )3!&is 'pply the matri" to reflect a point (-* /0 in the /2'"is

Example 2(

"ind the image of 2>%0&? under reflection in the y- axis.Solution


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"rom *y >x.y?9 >-x0y?

*y >% 0& ? 9> -%0&?

Therefore the image of 2>%0&? is 3849:!5 .

Refle%tion in t&e line y ' #.

The line y9x makes an angle &)( with x and y axes. 8t is the line of symmetry for theangle ZO[ formed by two axis. 2y using isosceles triangle properties0 reflection of thepoint >'0(? in the line y9x will be > (0'? while the reflection of >(0+? in the line y9xwill be > +0 (? it can be noticed that the coordinates are exchanging positions. Bencethe reflection of the point >x0y? in the line y9x is > y0x?.

here *y 9xmeans reflection in the line y9x.

Example 2*

"ind the image of the point $>'0+? after reflection in the line y 9 x . :raw a sketch.


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Refle%tion in t&e line y ' -#

The reflection of the point 2>x0y? in the line y 9 -x is 2N>-y0-x?.

Example 2+

"ind the image of 2 >%0&? after reflection in the line y9-x followed by another

reflection in the line y9(.:raw a sketch.#olutionM

Xeflection of 2 in the line y9-x is 2N>-&0-%?. The line y9( is the x @ axis. #o reflection >-&0-%? in the x-axis is >-&0%?

Therefore the image of 2 >%0&? is 3;49!:5.

The image of a point P (x,y) when reflected in the line making an angle α with positive x-axis

and passing through the origin.


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8f the line passes through the origin and makes an angle a with x @ axis in the positivedirection0 then its equation is y9 xtan\ where tan\is the slope of the line.

6onsider the following diagram.

2ut OA] is a right angled triangle.


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#o x 9 OA 6os^ and y 9 OA#in^ .

$gain OAYX is a right angled triangle and the angle AY]X 9 a -^ 7 a- ^7 ^0 this is dueto the fact that reflection is an isometric mapping.


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Example 2/

"ind the image of the point $ >'0 +? after a reflection in the line y 9 x.

Example 20


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"ind the image of 2 >%0&? after reflection in the line y 9 -x followed by another

reflection in the line y 9 (.

2ut the line y 9 ( has ( slope because it is the x @ axis0

Example

"ind the equation of the line y 9 +x 7 ) after being reflected in the line y 9 x0Solution

The line y 9 x has a slope '

#o tan a 9 ' which means a 9 &)(

To find the image of the line y 9 +x 7 )0 we choose at least two points on it and findtheir images0 then we use the image points to find the equation of the image line.


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The points >(0)? and >'0D? lie on the line

#o the image line is the line passing through >)0(? and >D0'? and it is obtained asfollowsM

Exe"#ise (

#elf Aractice.

(. ?ind the image of the point D &4,)' under reOection in the x – axis


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). >oint P &04,3' is reOected in the y – axis. ?ind its image coordinates.

3.


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"ind the image of the point A>'0(? after a rotation through ;( ( about the origin in the

anti clockwise direction.

A is on the x @ axis0 so after rotation through ;(( about the origin it will be on the y @axis. #ince A is 'unit from O0 AY is also ' unit from O0 the coordinates of AY >(0'? areAY >(0'?. Therefore X ;(

(>'0(? 9 >(0'?.

Example 2

"ind the image of the point 2 >&0+? after a rotation through ;( ( about the origin in the

anticlockwise direction.Solution

6onsider the following figure0


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Exe"#ise *

"ind the matrix of rotation through

a. Q22 about the origin

b. 4%2 about the origin

c. )*22 about the origin

"ind the image of the point >'0+? under rotation through '( ( ant @clockwise about the

origin.

"ind the image of the point >-+0'? under rotation through +D(( clockwise about the

origin

"ind the image of >'0+? after rotation of -;((.

"ind the image of the line passing through points a >-+0%? and 2>+0? after rotation

through ;(( clockwise about the origin


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eneral for"ula for rotation

6onsider the following sketch0


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Example

"ind the image of the point >'0+? under a rotation through '( ( anticlockwise

Therefore the image of >'0 +? after rotation through '( ( anticlockwise is >-'0-+?.

Example !

"ind the image of the point >)0+? under rotation of ;( ( followed by another rotation of

'(( anticlockwise.Solution


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Therefore the image of >)0+? under rotation of ;( ( followed by another rotation of'(( anticlockwise is >+0-)? .

T"anslation

Definition: $ translation is a mapping of a point A >x0 y? into AN >xN0 yN? by the `ector

>a0 b? such that >xN0 yN? 9 >x0 y? 7 >a0 b?0 translation is denoted by the letter T. #o Tmaps a point >x0 y? into xN0 yN?

here >xN0 yN? 9 >x0 y? 7 >a0 b?

6onsider the triangle OA] whose vertices are >(0(?0 >%0'? and >%0(? respectively whichis mapped into triangle OYAY]Y by moving it + units in the positive x direction and %units in the positive y direction


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Example (

8f T is a translation by the vector >&0%?0 find the image of >'0 +? under this translation.

Example *

$ translation T maps the point >-%0 +? into >&0 %?. "ind where >a? T maps the origin >b?

T maps the point >D0 &?.


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Example +

"ind the translation vector which maps the point >,0-,? into >D0',?.Solution

!iven that >x0 y? 9 >,0-,? and >xY0 yY? 9 >D0',?0 >a0 b? 9/

"rom T >x0 y? 9 >x0 y? 7 >a0 b? 9 >xN0 yN?0

then >D0',? 9 >,0-,?7>a0b? which means a9D-, 9 ' and b9',7, 9 ++. Thereforetranslation vector (a*b) ' (+*,,).

The Enlar$ement %atri& E in Enlar$in$ i$ures3se the enlargement matri" ; in enlarging figures

7e'inition Gnlargement is the transformation which magnifies an obEect such that itsimage is proportionally increases on decreased in size by some factor k. The generalmatrix of enlargement

Example /


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"ind the image of the square with vertices O>(0(?0 $ >'0(?0 2 >'0'? and 6 >(0'? under

the

Example 0


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"ind the image of >,0 ;? under enlargement by the matrix

Example !

:raw the image of a unit circle with center O >(0(? under

(0%?0 >%0(?0 >(0-%?0 >-%0(? and other pointsrespectively0 where the centre remains >(0(? and the radius becomes % units.


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8 n the figure above0 the circle with radius ' unit and its image with radius % units6'and 6+ respectively are shown.

inear Transfor"ation:

7e'inition

"or any transformation T0 any two vectors C and ` and any real number t0 T is said tobe a linear transformation if and only if

T(t ) ' tT() and T (/0) ' T() / T(0)

Example !1

#how that the rotation by ;((about O>(0(? is a linear trans formationSolution

Let C9>C'0C+? and ` 9>`' 0 `+? be any two vectors in the plane and t be any realnumber

To show that X;(( is the linear transformation we must show that

X;(( >tC?9 t X;(( >C? and

X;(( >C 7 `? 9 X;(( >C? 7 X;(( >`?


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Therefore0 since X;(( >C? 7 X;(( >`? 9 X;(( >C7`? and X;(( >tC?9 t X;(( >C?0 then X;((isa linear trans formation.

Example !2

#uppose that T is a linear transformation such thatT>C? 9 >'0-+?0 T>`? 9 >-%0-'? for any vectors C and `0 find

>a? T>C7 `? >b? T>C? >c? T>%C -+`?

Solution

>a?#ince T is a linear Transformation then

T> C7 `? 9 T>C? 7 T>`?


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Exe"#ise +

1. 8f

+. 8s the matrix of reflection in a line inclined at angle a0 C9>,0'? 0 `9>-'0&? and

a'%)((0 find >a? m>C7`? >b? m>+`?

8f C 9>+0-D? and `9>+0-%?0 find the matrix of linear transformation T such that T>+C?9>-

&0'&? and T>%`? 9 >,0;?

&. hat is the image of >'0+? under the transformation

). !iven that 8 is the identify transformation such that 8>C? 9C for any `ector C0 prove

that 8 is a linear transformation.

T#IG,O%ET#)

Trigonometry is a branch of mathematics that deals with relationship (s) between angles and sides of

triangles.


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Tri$onometric #atios

The Sine0 "osine and Tan$ent of an !n$le %easured in the

"loc


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8f _is an obtuse angle >;((3_3'((? then the trigonometrical ratios are the same as thetrigonometrical ratio of '((-_

8f _is a reflex angle >'((3 _3+D((? then the trigonometrical ratios are the same asthat of _- '((


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8f _is a reflex angle >+D((3 _3 %,((?0 then the trigonometrical ratios are the same as

that of %,(( -_


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e have seen that trigonometrical ratios are positive or negative depending on thesize of the angle and the quadrant in which it is found.

The result can be summarized by using the following diagram.


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Tri$onometric #atios to Solve Problems in +aily Life 'pply trigonometric ratios to solve problems in daily life

Example 1

rite the signs of the following ratios

a. :in (*22

b. #os )422

c. Tan 3(22

d. sin 322

Solution

a?#in 'D((

#ince 'D(( is in the second quadrant0 then #in 'D(( 9 #in >'((-'D((? 9 #in '((

R#in 'D(( 9 #in '((

b? 6os +&(( 9 -6os >+&((-'((?9 -6os ,((

Therefore 6os +&((9 -6os ,((

c? Tan %'(( 9 -Tan >%,((-%'((? 9 - Tan )((

Therefore Tan %'((9 -Tan )((

d? #in %(((9 -sin >%,((-%(((? 9 -sin ,((


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Therefore sin %(((9 - #in ,((

Relations$ip )et,een T"i&onomet"i#al "atios

The above relationship shows that the #ine of angle is equal to the cosine of itscomplement.

$lso from the triangle $26 above

$gain using the $26

b+ 9 a+7c+ >Aythagoras theorem?

$nd


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Example 2

!iven that$ is an acute angle and 6os $9 (.0 find

a. :in 5

b. tan 5.


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Example

8f $ and 2 are complementary angles0

Solution

8f $ and 2 are complementary angle

Then #in $ 9 6os 2 and #in 2 9 6os $

Example !

!iven that _and ^are acute angles such that _7 ^9 ;(( and #in_9 (.,0 find tan^Solution


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Exe"#ise 1

"or practice


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Sine and "osine unctionsSines and "osines of !n$les 5 Such That 39-56=S> 9-56

Find sines and cosines of angles 5 such that 28)5< = 8)5ᶿPositi-e an% Ne&ati-e an&les

$n angle can be either positive or negative.

Definition:

Positi-e an&le is an angle measures in anticlockwise direction from the positive [-axis

Ne&ati-e an&le is an angle measured in clockwise direction from the positive [-axis

Fa#ts


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a. ?rom the above ;gure if is a positive angle then the correspondingnegative angle to is &0 3922' or & 0 3922'

b. .If is a negative angle, its corresponding positive angle is &392'

Example (

"ind thecorresponding negative angle to the angle _if M

a. $ %7

2

b. $ )4%2

Example *


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hat is the positive angle corresponding to - &,/

S1E2I3 3NES

The angles included in this group are ((0 %((0 &)(0 ,((0 ;((0 '((0 +D((0 and %,((

2ecause the angle ((0 ;((0 '((0 +D((0 and %,((0 lie on the axes thentheirtrigonometrical ratios are summarized in the following table.

The $26 is an equilateral triangle of side + units


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"or the angle &)( consider the following triangle

The following table summarizes the 6osine0 #ine0 and tangent of the angle %( ( 0&)( and ,((


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NB: The following figure is helpful to remember the trigonometrical ratios of special

angles from (to ;(

8f we need the sines of the above given angles for examples0 we only need to take thesquare root of the number below the given angle and then the result is divided by +.


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Example +

"ind the sine0 cosine and tangents of each of the following angles

a. 0(3%2

b. ()22

c. 3322


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Example /

"ind the value of _if 6os _9 - and _5 _5 %,(Solution

#ince 6os _is @ >ve?0 then _lies in either the second or third quadrants0

'( @_9 - 6os >_7'((? 9 -9 -6os,((

#o _9 '((-,(( 9 '+(( or _9 '(( 7 ,(( 9 +&((

_9 '+(( (r _9+&((

Example 0

6onsider below

Exe"#ise 2

#olve the "ollowing.


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The Graphs of Sine and "osine#raw the graphs of sine and cosine

6onsider the following table of value for y9sin_ where _ranges from - %,(to %,(


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"or cosine consider the following table of values

"rom the graphs for the two functions a reader can notice that sin_and cos_both lie inthe interval -' and ' inclusively0 that is -'5sin_' and -'5cos_5' for all values of _.

The graph of y9 tan_is left for the reader as an exercise

NB: -45 tan654t&e sy"bol 4"eans infinite

$lso you can observe that both #in_nd cos_repeat themselves at the interval of

%,(0 which means sin_9 sin>_7%,(? 9 sin>_7+x%,((? etc

and 6os_9>6os_7%,((?9 6os>_7+x%,((?

Gach of these functions is called a period function with a period %,((

'. Csingtrigonometrical graphs in the interval -%,((5_5%,((

"ind _such that

a. :in$ 2.4

b. #os$ 2.Q

solution


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Example 1

Cse the graph of sin_to find the value of_if in_9 -'. and -%,(( 5_5%,((

Solution

in_9 -'.

#in_9 -'.& 9 -(.&)

#in_9 -(.&)

#o _9 -')%(0 -+D(0 +(D(0 %%%(

The $raphs of sine and cosine functions%nterpret the graphs of sine and cosine functions

Example 11

Cse thetrigonometrical function graphs for sine and cosine to find the value of

a. :in &0422'

b. #os &0422'Solution

a. :in &0422'$ 0 2.94

b. #os &0422'$ 2.*9

Sine and "osine #ulesThe Sine and "osine #ules

#erive the sine and cosine rules

6onsider the triangle $26 drawn on a coordinate plane


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"rom the figure above the coordinates of $0 2 and 6 are >(0 (?0 >c0 (? and >b6os_0b#in_? respectively.


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#8


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"ind the unknown sides and angle in a triangle $26 in which a9 ++.+cm29 ,and $9+,

Solution

2y sine rule

#in $9 sin 29 #in 6

Example 1

"ind unknown sides and angles in triangle $26

here a9%cm0 c9 &cm and 29 %(Solution

2y cosine rule0


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Example 1!

"ind the unknown angles in the following triangle


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Exe"#ise

'. !iven thata9''cm0 b9'&cm and c9+'cm0 "ind the Largest angle of $26

+. 8f $26: is a parallelogram whose sides are '+cm and ',cm what is the length of the

diagonal $6 if angle 29'';/


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%.8f $ and 2 are two ports on a straight 6oast line such that 2 is )%km east of $. $ ship

starting from $ sails &(km to a point 6 in a direction G,)


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"rom the figure above 32$:9\and 3$269^thus326:9\7^

"rom 26:


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"or 6os>\^? 6onsider the following unit circle with points A and ] on it such thatOA0makes angle\ with positive x-axis and O] makes angle ^with positive x-axes.


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"rom the figure above the distance d is given by


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8n general


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Example 1(

'. ithoutusing tables find the value of each of the following

a. :in *%U

b. #os(2%Solution:


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Example 1*

"ind

a. :in(%2U

b. #os (%U

Exe"#ise !

'. ithout using tables0 find


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a. :in(%U

b. #os ()2U

+. "ind #in ++) from >'(7&)?

%. `erify that

a. :inQ2U $ ( by using the fact that Q2U$4%U4%U

b. #osQ2U$2 by using the fact that Q2U$32U92U

&. Gxpress each of the following in terms of sine0 cosine and tangent of acute angles.

a. :in(2*U

b. #os322U

). 2y using the formula for #in >$-2?0 show that #in >;(-6?96os 6

'E"TO#S+isplacement and Positions of 'ectorsThe "oncept of a 'ector ?uantity

;"plain the concept of a vector quantity $ vector - is a physical quantity which has both magnitude and direction.

The +ifference (et*een +isplacement and Position 'ectors#istinguish between displacement and position vectors

8f an obEect moves from point $ to another point say 20 there is a displacement


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There are many `ector quantities0 some of which are displacement0 velocity0

acceleration0 force0 momentum0 electric field and magnetic field.

Other physical quantities have only magnitude0 these quantities are called S#ala"s.

"or example distance0 speed0 pressure0 time and temperature

Namin& o'


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Therefore two or more vectors are said to be equivalent if and only if they have samemagnitude and direction.

Position


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Components o' position -e#to"s

Example 1

rite the position vectors of the following points >a? $ >'0-'?0 >b? 2 >-&0-%?>c? 69 >u0 v? where C and ` are any real numbers and give their horizontal andvertically components


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Example 2

"or each of vectors a and b shown in figure below draw a pair of equivalent vectors

Solution:

The following figure shows the vectors a and ) and their respective pairs ofequivalent vectors


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!ny 'ector into I and @ "omponents

>esolving any vector into % and ? componentsT$e unit


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Example

rite the following vectors in terms of i and E vectors

Example !rite the following vectors as position vectors.


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%a$nitude and +irection of a 'ectorThe %a$nitude and +irection of a 'ector

,alculate the magnitude and direction of a vector Ma&nitu%e 4Mo%ules5 o' a


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- % 0 &?

hat is the magnitude of the vector C if C 9 &i @ )E/


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2earings are angles from a fixed direction in order to locate the interested places onthe earthNs surface.

Rea%in& )ea"in&s There are two method used to read bearings0 in the first methodall angles are measured with reference from the


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"rom figure above0 the direction of point $ from O is < &,( G 0 that of 2 is


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Csing the cosine rule

The displacement from *akambakoto *ikumiis '+ +,km2y sine rule


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$lternatively by using the scale $2 is approximately'&.% cm Therefore $2 9 '&.%x +(km 9 +,km and the bearing is obtained a protractor which is about


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Exe"#ise 1

1. "ind the magnitude of

Sum and Difference of Vectors

The Sum of T*o or %ore 'ectors


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Find the sum of two or more vectorsA%%ition o' -e#to"s

The sum of any two or more vectors is called the "esultant of the given vectors. Thesum of vectors is governed by triangle0 parallelogram and polygon laws of vectoraddition.

>1? Triangle law of vector $ddition

$dding two vectors involves Eoining two vectors such that the initial point of thesecond vector is the end point of first vector and the resultant is obtained bycompleting the triangle with the vector whose initial point is the initial point of thefirst vector and whose end points the end point of the second vector.

"rom the figure above a 7 ) is the resultant of vectors a and ) as shown below

>2? T$e pa"allelo&"am la,

hen two vectors have a common initial point say A0 then their resultant is obtainedby completing a parallelogram0 where the two vectors are the sides of the diagonalthrough A and with initial point at A


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Example 0

"ind the resultant of vectorsuand-in the following figure.

Solution

To get the resultant of vectors u and -0 you need to complete the parallelogram asshown in the following figure

"rom the figure above0 the result of u and - is AX 9 ? @


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Solution

8n the figure above A is the initial point of a: ) has been Eoined toaat point ] and # isEoined to ) at X0 while % is Eoined to # at point # and AT 9 a @ ) @ # @ % which is theresultant of the four vectors.

Opposite -e#to"s

Two vectors are said to be opposite to each other if they have the same magnitude

but different directions


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"rom the figure above a and ) have the same magnitude >%m? but opposite direction.

#o a and ) are opposite vectors.

Opposite vectors have zero resultant that is if a and ) are opposite vectors0 then

Example 11

"ind the vector p opposite to the vector " 9 ,i @ +E

The +ifference of 'ectorsFind the difference of vectors


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! 'ector by a Scalar+ultiply a vector by a scalar

8f a vector C has a magnitude m units and makes an angle_with a positive x axis0 then

doubling the magnitude of C gives a vector with magnitude +m.


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!enerally if C 9 >u'0 u+? and t is any non zero real number while >u'0 u+? are also realnumbers0 then

Example 12

8f a 9 %i 7 %E and ) 9 )i 7 &E"ind @ )a 7 %)


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Example 1

!iven that p 9 >0 ,? and q 9 >D0 ;?. "ind ;p @ q

!pplication of 'ectors

'ectors in Solvin$ Simple Problems on 'elocities0 +isplacementsand orces 'pply vectors in solving simple problems on velocities displacements and forces

`ector knowledge is applicable in solving many practical problems as in the followingexamples.

$ student walks &( m in the direction # &)( G from the dormitory to the parade groundand then he walks '((m due east to his classroom. "ind his displacement fromdormitory to the classroom.

Solution

6onsider the following figure describing the displacement which Eoins the dormitory:. parade ground A and 6lassroom 6.


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"rom the figure above the resultant is :6. 2y cosine rule

Example 1!

Three forces "' 9 >%0&?0 "+ 9 >)0-+? and "% 9 >&0%? measured in (0(?

a. Determine the magnitude and direction of their resultant.

b. #alculate the magnitude and direction of the opposite of the resultantforce.


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>b? Let the force opposite to " be "o0 then "o 9 -" 9 - >'+0 )? 9 >-'+0 -)?

"o9 '%< and its bearing is >,D.&(7'((? 9 +&D.&(

#o the magnitude and direction of the force opposite to the resultant force is '%< and#,D.&( respectively..

Exe"#ise 2

'. !iven that C 9 >%0 -&?0 `9 >-&0 %? and 9 >'0 '?0 calculate.a. The resultant of = V H

b. The magnitude and direction of the resultant calculated in part &a'above.

+. $ boat moves with a velocity of '(kmJh upstream against a downstream current of

'(kmJh. 6alculate the velocity of the boat when moving down steam.

%. Two forces acting at a point O makes angles of %(( and '%)( with their resultant

having magnitude +(< as shown in the diagram below.


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6alculate the magnitude and direction of the resultant of the velocities ` '9)i 7 ;E0`+ 9

&i 7 ,E and `% 9 &i @ %E where i and E are unit vectors of magnitude 'mJs in the positive

directions of the x and y axis respectively.

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