Additional Mathematics Project Work 2014 (Sabah)

8/12/2019 Additional Mathematics Project Work 2014 (Sabah)

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Additional Math Project Work 2014

APPLICATION OF VECTOR

Name : Shelyn Hiew Sam Tien

I/C No. : 970922-12-XXXX

Class : 5D


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Content

Content Page

Part 1 1

Part 2 4

Part 3 10

Further Exploration 13

Conclusion 15

Reflection 16


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PartA vector is a quantity that has both magnitude and direction. We can represent a

vector as a directed line segment as shown in Diagram 1, where the length is the

magnitude of the vector, and the arrow indicates the direction. The direction of thevector is from its tail to its head.

Tail

Head


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1. Write briefly the 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 Mechanics of Heron (first

century A.D.) of Alexandria. It was also the first corollary inIsaac Newtons(1642--1727) PrincipiaMathematica(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. Caspar Wessel (1745--1818), Jean Robert Argand (1768--

1822),Carl Friedrich Gauss (1777--1855), and at least one or two others conceived of complex

numbers as points in the two-dimensional plane, i.e., as two-dimensional vectors.

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. After a good deal of frustration, Hamilton was finally inspired to give up the search for such a

three-dimensional "number" system and instead he invented a four-dimensional system that he

called quaternions. At about the same time that Hamilton discovered quaternions,Hermann

Grassmann (1809--1877) was composing The Calculus of Extension (1844), now well known by its

German title,Ausdehnungslehre.During the middle of the nineteenth century, Benjamin Peirce (1809--1880) was far and away the most prominent mathematician in the United States, and he referred to

Hamilton as, "the monumental author of quaternions.

James Clerk Maxwell (1831--1879) was a discerning and critical proponent of

quaternions. William Kingdon Clifford (1845--1879) expressed "profound admiration" for

GrassmannsAusdehnungslehreand clearly favored vectors, which he often called steps, over

quaternions. The development of the algebra of vectors and of vector analysis as we know it today

was first revealed in sets of remarkable notes made byJ. Willard Gibbs (1839--1903) for his students

at Yale University. Vectors are now the modern language of a great deal of physics and applied

mathematics and they continue to hold their own intrinsic mathematical interest.
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2. List down 5 vector quantities.

Velocity, Acceleration, Momentum, Displacement, Force

3. Explain briefly, a situation that involves the application of vectors.

Ants use a mechanism calledpath integration, which requires them to measure distances and

direction. As an illustration, look at the zig-zag path shown in blue in figure 1, representing a foraging

path made up of several segments, starting from the nest at point Each segment is

represented by a vector, for example the first segment is given by the vector The sum of

the first vectors gives you a vector which points from the origin to the current location. The

negative of the vector points from the current location straight back to the nest. So to know your way

back home, you don't need to remember all the vectors you travelled along you simply add the

current one to the last total and take the negative.

Ants can approximate distances by counting their steps and use the position of the Sun as a

compass to keep track of the direction of each segment of an outward foraging route. Through

evolution, ants have developed neural circuits in their brain which can take information about distance

and direction and produce an output which is an approximation of the appropriate vector maths. The

result is a continuously updated home vector. The ants use this information when they have found

food or if they are attacked by a predator and need to get back home quick.


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Part 2A Boeing 737 aircraft maintains a constant velocity of 800 kilometers per hour due

South. The velocity of the jet stream is 100 kilometers 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.

(Please refer to graph 1a)

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.

Let Boeingbe the vector of Boeing 737 aircraft and jet stream be the vector jet stream.Angle of Boeing= 270Angle of jet stream= 45Magnitude of Boeing, = 800kmh

-1

Magnitude of jet stream, = 100kmh-1


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b) Express each vector above in the form = + and = .Use exact values (surds) for each vector and show your working.

Boeing= -80or Boeing= ( ) |= 100 = 100 = , since it is Northeasterly direction

= , hence

= 5000 = = 5000 =

jet stream= or jet stream=


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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) above. This is called the resultant vector, .a) Would you use the Triangle Law or the Parallelogram Law to find this sum?

Explain your choice.

Triangle Law. In the triangle law, the second vector is used when starting at the terminus

of the first vector and then join the base of the first vector to the terminus of the second

vector, thereby completing a triangle, to obtain the resultant vector.b) i) Based on your choice in 3(a), draw the resultant vector, , by using a

suitable scale.

(Please refer to graph 3b)

ii) Hence, find the magnitude and direction of the resultant vector from 3(b)

above.

732.71kmh-1

The direction of the resultant vector, , from the positive -axis is 275.5.

4. a) By using another method, find the magnitude of the resultant vector, .Show your working.

Resultant vector, R = Boeing + jet stream R = R = ( )

= = 732.71km-1


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b) Find the bearing of the resultant vector, , in degrees. Give your answer correct to one

decimal place.

=

= x 100BAC = 5.54

The direction of resultant vector, = from the positive -axis is 270+ 5.54= 275.54


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Part 3An aircraft is hkm above the ground at point P when it starts to land on point A with

angle of depression of 39

.


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c) If the aircraft eventually lands on point A within the range 7 8 minutes, what is

the range of the values of h? Give your answers correct to two decimal places.

Distance BA at 7 minutes = 115.81 x

= 13.51km

= BP = x 13.51

= 8.50 km

Distance BA at 8 minutes = 115.81 x

= 15.44km

= BP = x 15.44

= 9.72km

Hence, 8.50km BP 9.72km


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FURTHER EXPLORATION


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Magnitude = = = 81.85 km/h

=

= x 90

= 72.22

180- 60- 72.22

Direction = S(180- 60- 72.22)E

= S47.78E


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Conclusion

Vectors can be directed due East, due West, due South, and due North. But some vectors are

directed northeast (at a 45 degree angle); and some vectors are even directed northeast, yet more

north than east. Thus, there is a clear need for some form of a convention for identifying the direction

of a vector that is not due East, due West, due South, or due North. There are a variety of

conventions for describing the direction of any vector.

In conclusion, vectors can be represented by use of a scaled vector diagram. On such a

diagram, a vector arrow is drawn to represent the vector. The arrow has an obvious tail and

arrowhead. The magnitude of a vector is represented by the length of the arrow. A scale is indicated

(such as, 1 cm = 5 miles) and the arrow is drawn the proper length according to the chosen scale.

The arrow points in the precise direction. Directions are described by the use of some convention.

The most common convention is that the direction of a vector is the counter clockwise angle of

rotation which that vector makes with respect to due East.


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Reflection

In this additional mathematics project work, a lot of information that I found. I have learnt how vectors

appear in our daily life. Apart from that, this project encourages the student to work together and

share their knowledge. It is also encourage student to gather information from the internet, improve

thinking skills and promote effective mathematical communication. Not only that, I

had learned some moral values that I practice.

This project had taught me to responsible on the works that are

given to me to be completed. This project also had made me felt more

confidence to do works and not to give easily when we could not find the solution for the question. I

also learned to be more discipline on time, which I was given

about one week to complete this project and pass up to my

teacher just in time.

Last but not least, I proposed this project should be

continue because it brings a lot of moral value to the student

and also test the students understanding in Additional Mathematics.

Additional Mathematics Project Work 2014 (Sabah)

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