formfour mather notes
TRANSCRIPT
-
8/16/2019 FormFour Mather Notes
1/148
-
8/16/2019 FormFour Mather Notes
2/148
Example 2
"ind the solution to the following system of simultaneous equations by graphical
method.
-
8/16/2019 FormFour Mather Notes
3/148
Simultaneous Equations GraphicallySolve simultaneous equations graphically
-
8/16/2019 FormFour Mather Notes
4/148
Example
#olve the following simultaneous equations graphically and check your solution by a
non-graphical method
-
8/16/2019 FormFour Mather Notes
5/148
Example !
"ind the solution to the following system of simultaneous equations by graphical
method.
-
8/16/2019 FormFour Mather Notes
6/148
Exe"#ise 1
-
8/16/2019 FormFour Mather Notes
7/148
"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 $which is above the line satisfy the relation y1) and those lying in the half plane 2which is below the given line0 satisfy the relation y3 ).
-
8/16/2019 FormFour Mather Notes
8/148
S$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.
-
8/16/2019 FormFour Mather Notes
9/148
-
8/16/2019 FormFour Mather Notes
10/148
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 +.
-
8/16/2019 FormFour Mather Notes
11/148
:etermine the solution set of the simultaneous inequalities y 7 x 4% and x-+y 5 ;.
-
8/16/2019 FormFour Mather Notes
12/148
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?
-
8/16/2019 FormFour Mather Notes
13/148
(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
-
8/16/2019 FormFour Mather Notes
14/148
$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.
-
8/16/2019 FormFour Mather Notes
15/148
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 +
-
8/16/2019 FormFour Mather Notes
16/148
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
-
8/16/2019 FormFour Mather Notes
17/148
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
-
8/16/2019 FormFour Mather Notes
18/148
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?
-
8/16/2019 FormFour Mather Notes
19/148
#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
-
8/16/2019 FormFour Mather Notes
20/148
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
-
8/16/2019 FormFour Mather Notes
21/148
# 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,.
-
8/16/2019 FormFour Mather Notes
22/148
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,.
-
8/16/2019 FormFour Mather Notes
23/148
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?
-
8/16/2019 FormFour Mather Notes
24/148
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.
-
8/16/2019 FormFour Mather Notes
25/148
Example /
hat is the probability of not getting an even number when a fair die is tossed/
-
8/16/2019 FormFour Mather Notes
26/148
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 '/
-
8/16/2019 FormFour Mather Notes
27/148
-
8/16/2019 FormFour Mather Notes
28/148
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.
-
8/16/2019 FormFour Mather Notes
29/148
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
-
8/16/2019 FormFour Mather Notes
30/148
Example 1
8n a family of % children what is the probability that
a. 5ll are girls
b. 5t least two are boys
-
8/16/2019 FormFour Mather Notes
31/148
Example 1!
Three coins are tossed simultaneously. "ind the probability that
a. 3 heads appear
b. ) tails and one head appear
-
8/16/2019 FormFour Mather Notes
32/148
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.
-
8/16/2019 FormFour Mather Notes
33/148
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.
-
8/16/2019 FormFour Mather Notes
34/148
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
-
8/16/2019 FormFour Mather Notes
35/148
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
-
8/16/2019 FormFour Mather Notes
36/148
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.
-
8/16/2019 FormFour Mather Notes
37/148
%!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.
-
8/16/2019 FormFour Mather Notes
38/148
Types of "atri%es:
The following are the common types of matrices-
-
8/16/2019 FormFour Mather Notes
39/148
%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
-
8/16/2019 FormFour Mather Notes
40/148
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
-
8/16/2019 FormFour Mather Notes
41/148
&. $ 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
-
8/16/2019 FormFour Mather Notes
42/148
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
-
8/16/2019 FormFour Mather Notes
43/148
Example +
!iven0
Solution$
-
8/16/2019 FormFour Mather Notes
44/148
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
-
8/16/2019 FormFour Mather Notes
45/148
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
-
8/16/2019 FormFour Mather Notes
46/148
"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
-
8/16/2019 FormFour Mather Notes
47/148
Example 1
"ind 6S: if
(roduct of a matri" and an identity matri":
-
8/16/2019 FormFour Mather Notes
48/148
-
8/16/2019 FormFour Mather Notes
49/148
%.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
-
8/16/2019 FormFour Mather Notes
50/148
Example 12
"ind
Example 1
6onsidering
Example 1!
"ind the value of x
-
8/16/2019 FormFour Mather Notes
51/148
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
-
8/16/2019 FormFour Mather Notes
52/148
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 $
-
8/16/2019 FormFour Mather Notes
53/148
#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
-
8/16/2019 FormFour Mather Notes
54/148
$lsoto get r and s0 the same procedure must be followed
$nd
-
8/16/2019 FormFour Mather Notes
55/148
-
8/16/2019 FormFour Mather Notes
56/148
-
8/16/2019 FormFour Mather Notes
57/148
Example 1+
hich of the following matrices have inverses/
-
8/16/2019 FormFour Mather Notes
58/148
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.
-
8/16/2019 FormFour Mather Notes
59/148
- . - %atri& to Solve Simultaneous Equations 'pply ) * ) matri" to solve simultaneous equations
#olving simultaneous equations by matrix method
-
8/16/2019 FormFour Mather Notes
60/148
Then 29 $-'S6
Example 1/
2y matrix method solve the following simultaneous equations
-
8/16/2019 FormFour Mather Notes
61/148
*ultiplying $-' an each side of the equation0 gives0
-
8/16/2019 FormFour Mather Notes
62/148
Example 10
#olve
-
8/16/2019 FormFour Mather Notes
63/148
*ultiplying $-' on each side of the equation gives0
-
8/16/2019 FormFour Mather Notes
64/148
Example 2
2y using matrix method solve the following simultaneous equations
*ultiplying $-' on each side of the equation gives0
-
8/16/2019 FormFour Mather Notes
65/148
6ramerHs Xule
#o
-
8/16/2019 FormFour Mather Notes
66/148
Example 21"ind
Example 22
2y using 6ramerHs rule
-
8/16/2019 FormFour Mather Notes
67/148
Example 2
2yusing 6ramerHs rule0
Exe"#ise !'. Cse the matrix method to solve the following systems of simultaneous equations.
-
8/16/2019 FormFour Mather Notes
68/148
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
-
8/16/2019 FormFour Mather Notes
69/148
$ 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
-
8/16/2019 FormFour Mather Notes
70/148
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
-
8/16/2019 FormFour Mather Notes
71/148
"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.
-
8/16/2019 FormFour Mather Notes
72/148
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.
-
8/16/2019 FormFour Mather Notes
73/148
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.
-
8/16/2019 FormFour Mather Notes
74/148
#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.
-
8/16/2019 FormFour Mather Notes
75/148
Example 2/
"ind the image of the point $ >'0 +? after a reflection in the line y 9 x.
Example 20
-
8/16/2019 FormFour Mather Notes
76/148
"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.
-
8/16/2019 FormFour Mather Notes
77/148
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
-
8/16/2019 FormFour Mather Notes
78/148
). >oint P &04,3' is reOected in the y – axis. ?ind its image coordinates.
3.
-
8/16/2019 FormFour Mather Notes
79/148
"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
-
8/16/2019 FormFour Mather Notes
80/148
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
-
8/16/2019 FormFour Mather Notes
81/148
eneral for"ula for rotation
6onsider the following sketch0
-
8/16/2019 FormFour Mather Notes
82/148
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
-
8/16/2019 FormFour Mather Notes
83/148
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
-
8/16/2019 FormFour Mather Notes
84/148
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 &?.
-
8/16/2019 FormFour Mather Notes
85/148
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 /
-
8/16/2019 FormFour Mather Notes
86/148
"ind the image of the square with vertices O>(0(?0 $ >'0(?0 2 >'0'? and 6 >(0'? under
the
Example 0
-
8/16/2019 FormFour Mather Notes
87/148
"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.
-
8/16/2019 FormFour Mather Notes
88/148
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;(( >`?
-
8/16/2019 FormFour Mather Notes
89/148
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>`?
-
8/16/2019 FormFour Mather Notes
90/148
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.
-
8/16/2019 FormFour Mather Notes
91/148
Tri$onometric #atios
The Sine0 "osine and Tan$ent of an !n$le %easured in the
"loc
-
8/16/2019 FormFour Mather Notes
92/148
8f _is an obtuse angle >;((3_3'((? then the trigonometrical ratios are the same as thetrigonometrical ratio of '((-_
8f _is a reflex angle >'((3 _3+D((? then the trigonometrical ratios are the same asthat of _- '((
-
8/16/2019 FormFour Mather Notes
93/148
8f _is a reflex angle >+D((3 _3 %,((?0 then the trigonometrical ratios are the same as
that of %,(( -_
-
8/16/2019 FormFour Mather Notes
94/148
e have seen that trigonometrical ratios are positive or negative depending on thesize of the angle and the quadrant in which it is found.
The result can be summarized by using the following diagram.
-
8/16/2019 FormFour Mather Notes
95/148
Tri$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 ,((
-
8/16/2019 FormFour Mather Notes
96/148
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?
$nd
-
8/16/2019 FormFour Mather Notes
97/148
Example 2
!iven that$ is an acute angle and 6os $9 (.0 find
a. :in 5
b. tan 5.
-
8/16/2019 FormFour Mather Notes
98/148
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
-
8/16/2019 FormFour Mather Notes
99/148
Exe"#ise 1
"or practice
-
8/16/2019 FormFour Mather Notes
100/148
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
-
8/16/2019 FormFour Mather Notes
101/148
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 *
-
8/16/2019 FormFour Mather Notes
102/148
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
-
8/16/2019 FormFour Mather Notes
103/148
"or the angle &)( consider the following triangle
The following table summarizes the 6osine0 #ine0 and tangent of the angle %( ( 0&)( and ,((
-
8/16/2019 FormFour Mather Notes
104/148
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 +.
-
8/16/2019 FormFour Mather Notes
105/148
Example +
"ind the sine0 cosine and tangents of each of the following angles
a. 0(3%2
b. ()22
c. 3322
-
8/16/2019 FormFour Mather Notes
106/148
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.
-
8/16/2019 FormFour Mather Notes
107/148
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 %,(
-
8/16/2019 FormFour Mather Notes
108/148
"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
-
8/16/2019 FormFour Mather Notes
109/148
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
-
8/16/2019 FormFour Mather Notes
110/148
"rom the figure above the coordinates of $0 2 and 6 are >(0 (?0 >c0 (? and >b6os_0b#in_? respectively.
-
8/16/2019 FormFour Mather Notes
111/148
#8
-
8/16/2019 FormFour Mather Notes
112/148
-
8/16/2019 FormFour Mather Notes
113/148
"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
-
8/16/2019 FormFour Mather Notes
114/148
-
8/16/2019 FormFour Mather Notes
115/148
Example 1!
"ind the unknown angles in the following triangle
-
8/16/2019 FormFour Mather Notes
116/148
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'';/
-
8/16/2019 FormFour Mather Notes
117/148
%.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,)
-
8/16/2019 FormFour Mather Notes
118/148
"rom the figure above 32$:9\and 3$269^thus326:9\7^
"rom 26:
-
8/16/2019 FormFour Mather Notes
119/148
"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.
-
8/16/2019 FormFour Mather Notes
120/148
"rom the figure above the distance d is given by
-
8/16/2019 FormFour Mather Notes
121/148
8n general
-
8/16/2019 FormFour Mather Notes
122/148
Example 1(
'. ithoutusing tables find the value of each of the following
a. :in *%U
b. #os(2%Solution:
-
8/16/2019 FormFour Mather Notes
123/148
Example 1*
"ind
a. :in(%2U
b. #os (%U
Exe"#ise !
'. ithout using tables0 find
-
8/16/2019 FormFour Mather Notes
124/148
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
-
8/16/2019 FormFour Mather Notes
125/148
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'
-
8/16/2019 FormFour Mather Notes
126/148
Therefore two or more vectors are said to be equivalent if and only if they have samemagnitude and direction.
Position
-
8/16/2019 FormFour Mather Notes
127/148
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
-
8/16/2019 FormFour Mather Notes
128/148
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
-
8/16/2019 FormFour Mather Notes
129/148
!ny 'ector into I and @ "omponents
>esolving any vector into % and ? componentsT$e unit
-
8/16/2019 FormFour Mather Notes
130/148
Example
rite the following vectors in terms of i and E vectors
Example !rite the following vectors as position vectors.
-
8/16/2019 FormFour Mather Notes
131/148
%a$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
-
8/16/2019 FormFour Mather Notes
132/148
- % 0 &?
hat is the magnitude of the vector C if C 9 &i @ )E/
-
8/16/2019 FormFour Mather Notes
133/148
-
8/16/2019 FormFour Mather Notes
134/148
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
-
8/16/2019 FormFour Mather Notes
135/148
"rom figure above0 the direction of point $ from O is < &,( G 0 that of 2 is
-
8/16/2019 FormFour Mather Notes
136/148
Csing the cosine rule
The displacement from *akambakoto *ikumiis '+ +,km2y sine rule
-
8/16/2019 FormFour Mather Notes
137/148
$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
-
8/16/2019 FormFour Mather Notes
138/148
Exe"#ise 1
1. "ind the magnitude of
Sum and Difference of Vectors
The Sum of T*o or %ore 'ectors
-
8/16/2019 FormFour Mather Notes
139/148
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
-
8/16/2019 FormFour Mather Notes
140/148
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 ? @
-
8/16/2019 FormFour Mather Notes
141/148
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
-
8/16/2019 FormFour Mather Notes
142/148
"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
-
8/16/2019 FormFour Mather Notes
143/148
! '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.
-
8/16/2019 FormFour Mather Notes
144/148
!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 %)
-
8/16/2019 FormFour Mather Notes
145/148
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.
-
8/16/2019 FormFour Mather Notes
146/148
"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.
-
8/16/2019 FormFour Mather Notes
147/148
>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.
-
8/16/2019 FormFour Mather Notes
148/148
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.