additional mathematics project spm

8/21/2019 Additional Mathematics Project SPM

1/28

SEKOLAH MENENGAH KEBANGSAAN

KINARUT, PETI SURAT 637,89608

PAPAR, SABAH.

Additional mathematics project

work

2014

Vectors applications

NAME : RENATHA JIFFRIN

I.C NUMBER : 970213-12-XXXX

CLASS: 5 HARMONI (2014)

TEACHERS NAME: EN. NOR ZAWARI HARON


2/28

contents

CONTENTS 1

TITLE 2

OBJECTIVES 3-4

FOREWARD 5

PART 1 6-11

PART 2 12-18

PART 3 19-22

FURTHER

EXPLORATION

23-24

CONCLUSION 25

REFLECTION 26


3/28

1

TITLE

VECTOR


4/28

2

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 problem-solving strategies then

to solve routine and non-routine 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 real-life

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 problem-solving which can increase interest and

confidence.

A step of preparation for the future undertakings and in

workplace.


5/28

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.

3

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.


6/28

4

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.


7/28

5

PART


8/28

6

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 (384--322 B.C.), and it is in the Mechanicsof Heron (firstcentury A.D.) of Alexandria. It was also the first corollary inIsaac

Newtons(1642--1727) 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.htm


9/28

7

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 (1831--1879) 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.htm


10/28

Thomson, 1824--1907) between the flow of heat and the distribution of

electrostatic forces.

8

William Kingdon Clifford

William Kingdon Clifford (1845--1879) 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 (1850--1925), a self-educated 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.htm


11/28

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)

9

A vector is formally defined as an element of avector space.In the

commonly encounteredvector space (i.e., Euclideann-space), 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 two-vector, an -dimensionalvector is often called ann-vector,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/n-Space.htmlhttp://mathworld.wolfram.com/n-Space.htmlhttp://mathworld.wolfram.com/n-Space.htmlhttp://mathworld.wolfram.com/n-Vector.htmlhttp://mathworld.wolfram.com/n-Vector.htmlhttp://mathworld.wolfram.com/n-Vector.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/n-Vector.htmlhttp://mathworld.wolfram.com/n-Space.htmlhttp://mathworld.wolfram.com/VectorSpace.htmlhttp://mathworld.wolfram.com/VectorSpace.html


12/28

10

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)


13/28

- 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

triangular-shaped direction of the crossing river process.

11

PART 2


14/28

12

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 x-axis. 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


15/28

13

ANSWER:

2(b)


16/28

14

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:


17/28

15

ANSWER:


18/28

16

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.


19/28

17

ANSWER:

4(a) & 4(b) :


20/28

18

PART 3


21/28

19

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:


22/28

The velocity of the aircraft when it descends from point Bto pointA is

115.808 kmh-1, assuming the velocity of the wind is 0 kmh-1.

20

(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 7-8minutes, 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 kmh-1

Distance of BA =

= 13.51kmh-1


23/28

*convert 7 minutes to seconds *115.808 is from AB

21

If the aircraft eventually lands on point A within 8 minutes, the

distance of BA is

Distance of BA =

= 15.44kmh-1


24/28

22

FURTHER EXPLORATION

ANSWER:


25/28

23


26/28

24

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.


27/28

25

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.


28/28

26

additional mathematics project spm

Documents