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8/21/2019 Additional Mathematics Project SPM
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SEKOLAH MENENGAH KEBANGSAAN
KINARUT, PETI SURAT 637,89608
PAPAR, SABAH.
Additional mathematics project
work
2014
Vectors applications
NAME : RENATHA JIFFRIN
I.C NUMBER : 97021312XXXX
CLASS: 5 HARMONI (2014)
TEACHERS NAME: EN. NOR ZAWARI HARON
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contents
CONTENTS 1
TITLE 2
OBJECTIVES 34
FOREWARD 5
PART 1 611
PART 2 1218
PART 3 1922
FURTHER
EXPLORATION
2324
CONCLUSION 25
REFLECTION 26
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1
TITLE
VECTOR
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objectivesThis project can be done in groups or individually but each of us
has to submit an individually written report. Upon the completion
of this Additional Mathematics Project Work we then will be able
to obtain valuable experiences and able to;
Apply and adapt varieties of problemsolving strategies then
to solve routine and nonroutine problems.
Experience classroom environment which is
challenging, interesting and meaningful hence improve our thinking
skills.
Experience the environments of the classroom where knowledge
and skills are applied in proper ways in solving reallife
problems
Experience classroom environments where expressing ones
mathematical thinking, reasoning and communication are highly
encouraged and expected.
Experience the environments that stimulate and enhance the
effective learning.
Acquire effective mathematical communication through oral and
writing, and to use mathematics language to express mathematical
ideas correctly and precisely. Enhance acquisition of mathematical knowledge and skills
through problemsolving which can increase interest and
confidence.
A step of preparation for the future undertakings and in
workplace.
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Realise that mathematics is an important and powerful
instrument in order to solve the problems in real life hence
sparking a positive respond toward mathematics.
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Able to train ourselves to collaborate, cooperate, and share
the knowledge toward the people surround us.
Use the technologies and ICT affectively.
Train ourselves to appreciate intrinsic values of
mathematics and to become more creative and innovative.
Realize the beauty and importance of mathematics formulasand solutions.
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FOREWARD
First of all, I would like to thank God, for giving me the strength
and health to do this project work. Not to forget my parents for providing
everything, such as money; to buy anything that are related to this projectwork. Moreover, their advice is very important to complete this project.
They also supported me and encouraged me to complete this task so that I
will not procrastinate in doing it. In addition, I would like to thank my
teacher, En. Nor Zawari Haron for the guidance to the whole class
throughout this project. We had some difficulties in doing this task, but he
has taught us patiently until we managed to complete this task. Last but not
least, thank you to my friends who has helped each other in order to
complete this task.
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PART
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1.HISTORY OF VECTORS
The parallelogram law for the addition of vectors is so intuitive that its
origin is unknown. It may have appeared in a now lost work
ofAristotle (384322 B.C.), and it is in the Mechanicsof Heron (firstcentury A.D.) of Alexandria. It was also the first corollary inIsaac
Newtons(16421727) Principia Mathematica(1687). In
the Principia,Newton dealt extensively with what are now considered
vectorial entities (e.g., velocity, force), but never the concept of a vector.
The systematic study and use of vectors were a 19thand early 20thcentury
phenomenon.Vectors were born in the first two decades of the 19thcentury
with the geometric representations of complex numbers.
August Ferdinand Mbius
In 1827, August Ferdinand Mbius published a short book, The Barycentric
Calculus, in which he introduced directed line segments that he denoted by
letters of the alphabet, vectors in all but the name. In his study of centers of
gravity and projective geometry, Mbius developed an arithmetic of these
directed line segments; he added them and he showed how to multiply
them by a real number. His interests were elsewhere, however, and no one
else bothered to notice the importance of these computations.
http://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/aristotle.htmhttp://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/newton.htmhttp://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/newton.htmhttp://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/newton.htmhttp://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/newton.htmhttp://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/newton.htmhttp://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/aristotle.htm8/21/2019 Additional Mathematics Project SPM
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William Rowan Hamilton
In 1837, William Rowan Hamilton (18051865) showed that the complex
numbers could be considered abstractly as ordered pairs (a,b) of real
numbers. This idea was a part of the campaign of many mathematicians,
including Hamilton himself, to search for a way to extend the two
dimensional numbers to three dimensions; but no one was able to
accomplish this, while preserving the basic algebraic properties of real and
complex numbers.
James Clerk Maxwell
James Clerk Maxwell (18311879) was a discerning and critical proponent
of quaternions. Maxwell and Tait were Scottish and had studied together in
Edinburgh and at Cambridge University, and they shared interests in
mathematical physics. In what he called "the mathematical classification of
physical quantities," Maxwell divided the variables of physics into two
categories, scalars and vectors. Then, in terms of this stratification, he
pointed out that using quaternions made transparent the mathematical
analogies in physics that had been discovered byLord Kelvin (Sir William
http://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/maxwell.htmhttp://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/maxwell.htmhttp://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/maxwell.htmhttp://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/thomson.htmhttp://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/thomson.htmhttp://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/maxwell.htmhttp://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/maxwell.htm8/21/2019 Additional Mathematics Project SPM
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Thomson, 18241907) between the flow of heat and the distribution of
electrostatic forces.
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William Kingdon Clifford
William Kingdon Clifford (18451879) expressed "profound admiration"
for GrassmannsAusdehnungslehreand clearly favored vectors, which he
often called steps, over quaternions. In his Elements of Dynamic(1878),
Clifford broke down the product of two quaternions into two very different
vector products, which he called the scalar product(now known as the dot
product) and the vector product(today we call it the cross product). Forvector analysis, he asserted "[M]y conviction [is] that its principles will
exert a vast influence upon the future of mathematical science." Though
the Elements of Dynamicwas supposed to have been the first of a sequence
of textbooks, Clifford never had the opportunity to pursue these ideas
because he died quite young.
Oliver Heaviside
Oliver Heaviside (18501925), a selfeducated physicist who was greatly
influenced by Maxwell, published papers and his Electromagnetic
Theory(three volumes, 1893, 1899, 1912) in which he attacked
quaternions and developed his own vector analysis. Heaviside had receivedcopies of Gibbss notes and he spoke very highly of them. In introducing
http://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/heaviside.htmhttp://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/heaviside.htmhttp://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/heaviside.htmhttp://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/heaviside.htm8/21/2019 Additional Mathematics Project SPM
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Maxwells theories of electricity and magnetism into Germany (1894),
vector methods were advocated and several books on vector analysis in
German followed. Vector methods were introduced into Italy (1887, 1888,
1897), Russia (1907), and the Netherlands (1903)
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A vector is formally defined as an element of avector space.In the
commonly encounteredvector space (i.e., Euclideannspace), a vector is
given by coordinates and can be specified as . Vectors are
sometimes referred to by the number of coordinates they have, so a 2
dimensional vector is often called a twovector, an dimensionalvector is often called annvector,and so on.
Vectors can be added together (vector addition), subtracted (vector
subtraction)and multiplied byscalars (scalar multiplication).Vector
multiplication is not uniquely defined, but a number of different types of
products, such as thedot product,cross product,andtensor direct
product can be defined for pairs of vectors.
Diagram 1.1(d)
Based on Diagram 1.1(d), a vector from a point to a point is denoted ,
and a vector may be denoted , or more commonly. The point is often called
the "tail" of the vector, and is called the vector's "head." A vector with unit
length is called aunit vector and is denoted using ahat, .
http://mathworld.wolfram.com/VectorSpace.htmlhttp://mathworld.wolfram.com/VectorSpace.htmlhttp://mathworld.wolfram.com/nSpace.htmlhttp://mathworld.wolfram.com/nSpace.htmlhttp://mathworld.wolfram.com/nSpace.htmlhttp://mathworld.wolfram.com/nVector.htmlhttp://mathworld.wolfram.com/nVector.htmlhttp://mathworld.wolfram.com/nVector.htmlhttp://mathworld.wolfram.com/VectorAddition.htmlhttp://mathworld.wolfram.com/VectorSubtraction.htmlhttp://mathworld.wolfram.com/VectorSubtraction.htmlhttp://mathworld.wolfram.com/Scalar.htmlhttp://mathworld.wolfram.com/ScalarMultiplication.htmlhttp://mathworld.wolfram.com/VectorMultiplication.htmlhttp://mathworld.wolfram.com/VectorMultiplication.htmlhttp://mathworld.wolfram.com/DotProduct.htmlhttp://mathworld.wolfram.com/CrossProduct.htmlhttp://mathworld.wolfram.com/TensorDirectProduct.htmlhttp://mathworld.wolfram.com/TensorDirectProduct.htmlhttp://mathworld.wolfram.com/UnitVector.htmlhttp://mathworld.wolfram.com/Hat.htmlhttp://mathworld.wolfram.com/Hat.htmlhttp://mathworld.wolfram.com/UnitVector.htmlhttp://mathworld.wolfram.com/TensorDirectProduct.htmlhttp://mathworld.wolfram.com/TensorDirectProduct.htmlhttp://mathworld.wolfram.com/CrossProduct.htmlhttp://mathworld.wolfram.com/DotProduct.htmlhttp://mathworld.wolfram.com/VectorMultiplication.htmlhttp://mathworld.wolfram.com/VectorMultiplication.htmlhttp://mathworld.wolfram.com/ScalarMultiplication.htmlhttp://mathworld.wolfram.com/Scalar.htmlhttp://mathworld.wolfram.com/VectorSubtraction.htmlhttp://mathworld.wolfram.com/VectorSubtraction.htmlhttp://mathworld.wolfram.com/VectorAddition.htmlhttp://mathworld.wolfram.com/nVector.htmlhttp://mathworld.wolfram.com/nSpace.htmlhttp://mathworld.wolfram.com/VectorSpace.htmlhttp://mathworld.wolfram.com/VectorSpace.html8/21/2019 Additional Mathematics Project SPM
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2. FIVE VECTOR QUANTITIES
A physical quantities which has both magnitude and direction.
Displacement: a vector quantity which represents the difference in
the position of two points. It is given the symbol s and has unit of
metres(m) in a specified direction.
Force : if a force is applied on an object, the object will accelerate in
proportion to the magnitude of the force and in the direction of the
applied force.
Acceleration: the rate of change of velocity.
Velocity : is the rate of change of displacement.
Momentum: a vector quantity which is defined as the product ofmass and velocity.
3. A SITUATION THAT INVOLVES THE APPLICATION OF THE
VECTORS
*When crossing a flowing river.
Diagram 1.1(e)
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 The start is the point where you were about to cross a flowing river.
 In this case, you need to know what point you will land on the
opposite bank.
 This situation only can be known through the application of vector.

Based on Diagram 1.1(e), it shows that vector is applied as there is a
triangularshaped direction of the crossing river process.
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PART 2
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A Boeing 737 aircraft maintains a constant velocity of 800 kilometres per
hour due South. The velocity of the jet is 100 kilometres per hour in the
Northeast direction.
1.
Sketch the given vectors, with initial points at the origin, as
accurately as possible on your graph paper. Scale your axes.
ANSWER: *in graph
2. a) Determine the angle ,, in degrees, for each vector measured in
an anticlockwise direction from the positive xaxis. Then, state the
magnitude of each vector.
b)Express each vector above in the form v = x i+ y j and v =(.
Use exact values(surds) for each vector and show your
working.
ANSWER:
2.a)  The angle , of the vector of the velocity of the aircraft= 270South
 The magnitude of the vector of the velocity of the aircraft = 800
 The angle ,of the vector of the velocity of the air = 45 South East
 The magnitude of the vector of the velocity of the air = 100
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ANSWER:
2(b)
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3. The actual course of the plane is the sum of the two given vectors
as stated in question 2(a). This is called the resultant vector, VR.
(a) Would you use the Triangle Law or the Parallelogram Law
to find this sum? Explain your chose.
ANSWER:
 I would choose the Triangle Law in order to find the sum of the two
vectors given because it is easy to draw the diagram of the Triangle
Law.
(b) i) Based on your choice in 3(a), draw the resultant vector, VR
by using a suitable scale.
ANSWER: *in graph.
ii) Hence, find the magnitude and direction of the resultant
vector from 3(b)(i)
ANSWER:
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ANSWER:
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4.
(a) by using another method, find the magnitude of the
resultant vector, VR . Show your working.
(b)
Find the bearing of the resultant vector, , in degrees. Give
your answer correct to one decimal place.
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ANSWER:
4(a) & 4(b) :
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PART 3
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An aircraft is hkm above the ground at point Pwhen it starts to land on
pointAwith angle of depression of 39.
(a) Calculate the velocity of the aircraft when it descends from point B
to pointA. State your assumption(s).
ANSWER:
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The velocity of the aircraft when it descends from point Bto pointA is
115.808 kmh1, assuming the velocity of the wind is 0 kmh1.
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(b) Based on Diagram 4 and your answer in PART 3(a), calculate the
horizontal component and the vertical component of vector
ANSWER:
(c)
If the aircraft eventually lands on point A within the range 78minutes, what is the range of the values of h? Give your answers
correct to two decimal places.
ANSWER:
(c)If the aircraft eventually land on point A within 7 minutes, the
distance of BA is 13.51 kmh1
Distance of BA =
= 13.51kmh1
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*convert 7 minutes to seconds *115.808 is from AB
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If the aircraft eventually lands on point A within 8 minutes, the
distance of BA is
Distance of BA =
= 15.44kmh1
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FURTHER EXPLORATION
ANSWER:
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CONCLUSION
By doing this project work, I can conclude that the theory of
Vector has both magnitude and direction. This theory can also be used to
calculate the distance from one point to another point.
Moreover, by conducting research on the history of vectors
and the applications of vectors in our daily life made me understand even
deeply of its importance to our real life situations; such as crossing a
flowing river, in sailing and the navigator of a plane.
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reflection
After doing research, answering questions, drawing graph, making
conjectures, conclusion and some problems solving, I had realized that
Additional Mathematics is vital in our daily life. Throughout this project
work, it is quite enjoyable and interesting project because this project
made me to plan things carefully and precisely in systematic condition. In
fact, the further exploration is a good session for me to applied the
situation when I am facing some problem solving in real life that will make
me to use my knowledge on vectors. In a nutshell, I barely can apply the
concepts and skills that I have learned in problem solving in Additional
Mathematics. For my opinion, this project work is very beneficial for all the
students in our country.
VISION WITHOUT ACTION IS A DAYDREAM. ACTION WITHOUT VISION IS
A NIGHTMARE JAPANESE PROVERB.
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