Anuja
Anuja
Anuja
ABSTRACT
The main objective of the study is to make students aware about the
vectors and its life applications. Some students are do not like mathematics for
the reason that they scared of it and do not feel like they are able to understand
the concept.
Vectors are used in science to describe anything that has both a direction
and a magnitude. For example translations, displacements, velocities, forces
etc. vectors define in this way are caked free vectors.
Also they acquire the thorough knowledge on the topic vector and its
real life applications. And thus they may develop the skill of possible solving
and observing the given situation suitable using vectors.
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INTRODUCTION
Vectors are used in science to describe anything that has both a direction
and a magnitude. They are usually drawn as pointed arrows, the length of
which represents the vector’s magnitude. A quarterback’s pass is a good
example, because it has a direction (usually somewhere down filed) and a
magnitude (how hard the ball is thrown).
Off the field, vectors can be used to represent any number of physical
objects or phenomina wind, for instance, is a vectorial quantity, because at any
given location it has a direction and a magnitude. You could make a map of
airflow at any point in time, then by drawing wind vectors for a number of
different geographic locations.
Many properties of moving objects are also vectors. Take, for instance, a
billiard ball rolling across a table. The balls velocity vector described its
movement. The direction of the vector arrow marks the ball’s direction of
motion, and the length of the vector represents the speed of the ball.
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Vectors are important in navigation where the actual velocity of an
aeroplane relative to the earth is given by the combine velocities of the wind
together with the velocity which the plane would have in still air.
Vectors is a mathematical object that has magnitude and direction. With
other words it is a line of given length and pointing along a given direction.
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OBJECTIVES
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HYPOTHESIS
I assume that this project done by me on the topic ‘vectors’ will help the
students to understand the basic concepts and different uses as well as its
applications. In various fields, I hope the students will be able to use and apply
the concepts of vectors for the development of themselves and for the society.
This could also enable the students to improve their problem solving
abilities as well as the critical thinking and cognitive powers.
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METHODS AND PROCEDURE
Definition of a Vector
A vector is an object that has both a magnitude and a direction.
Geometrically, we can picture a vector as a directed line segment, whose length
is the magnitude of the vector and with an arrow indicating the direction. The
direction of the vector is from its tail to its head.
Two vectors are the same if they have the same magnitude and direction.
This means that if we take a vector and translate it to a new position, then the
vector we obtain at the end of this process is the same vector we had in the
beginning.
Two examples of vectors are those that represent force and velocity.
Both force and velocity are in a particular direction. The magnitude of the
vector would indicate the strength of the force or the speed associated with the
velocity.
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Addition of Vectors
1. The commutative law, which status the order of addition dosen’t matter :
a+b = b+a
This Law is also called the parallelogram Law, as illustrated in the
below image. Two of the edges of the parallelogram define a+b, and the
other pair of edges define b+a. But, both sums are equal to the same
diagonal of the parallelogram.
2. The associative Law, which states that the sum of three vectors does not
depend on which pair of Vectors is added first.
(a+b)+c = a+ (b+c)
Vector Substraction
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We define the vector –a, which is the opposite of a. The vector – a is the
vector with the same magnitude as a but that is pointed in the opposite
direction.
a-b, then, is the same thing as a + (-b). For instance, Let’s take the two vectors
a and b:
Magnitude of a Vector
The Magnitude of a vector is shown by two vertical bars on either side
of the vector :
|a|
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Or it can be written with double vertical bars :
||a||
We use pythagora’s theorem to calculate it :
|a|=
Unit Vector
A unit vector is any vector with a length of one; normally unit vectors
are used simply to indicate direction. A vector of arbitrary length can be
divided by its length to create a unit vector. This is known as normalizing a
vector. A unit vector is often indicated with a hat as in â.
Zero Vector
The zero vector is the vector with length zero. Written out in
coordinates, the vector is (0,0,0), and it is commonly denoted © or simply o.
The sum of the zero vector with any vector a is a (ie, o+a=a).
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a= = = = +
ie, the sum of all vectors, a,b,c,d, is given by the single vector joining
the start of the first to the end of the last in theis case, . This follows
directly from our previous definition [Addition of vectors] of the sum of
two vectors.
Similarly :
+ + + = ………………………
To Equal Vectors
It two vectors, a and b, are said to be equal, they have the same
magnitude and the same direction.
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If a=b, then
(a) a=b (magnitude equal)
(b) the direction of a = direction of b, ie: the two vectors are parallel and in
the same sense.
EXAMPLES OF VECTORS
VECTOR APPLICATION
Addition:
When two vectors point in same direction, simply add them together.
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Example : A man walks 4 6.5 m east, then another 20m east. Calculate his
displacement relative to where he started.
Subtraction
When two vectors point in the opposite direction, simply subtract them.
Example : A man walks 46.5 east, then another 20m west calculate the
displacement relative to where he started.
VECTOR GRAPHICS
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Vector graphics refers to the use of geometrical primitives such as
points, lines and curves (ie, shapes based on mathematical equations) instead of
resolution dependent bitmap graphics to represent images in computer
graphics. In video games this type of projection is somewhat rare, but has
become more common in recent years in browser based gaming with the advent
to Flash, since Flash supports vector graphics natively..
Vector game can also refer to a video game that uses a victor graphics
display capable of projecting images using an electron beam to draw images
instead of with pixels, much like a laser show. Many early arcade games used
such displays, as they were capable of displaying more detailed images than
raster displays on the hardware available at that time many vector – based
arcade game used full colour overlays to complement the otherwise
monochrome vector images. Other uses of these overlays were very detailed
drawings of the static gaming environment, while the moving objects were
drawn by the vector beam.
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results from a graphic artist work is created and saved as a sequence of vector
statement.
VECTOR ART
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A photograph that you take with your digital camera is made up of
thousands of pixels (picture elements), each one a different colour. Such
photos are called raster images.
Another name given to raster image is bitmap, which means every pixel
has information associated with it, like color and brightness.
One of the big problems with raster images is that if you want to enlarge
the image, it ends up looking blotchy, like we see in the lower image. The
edges are what we call ‘pixelated’ and they are not attractive at all.
The following picture are from a vector image of a hot air balloon. The digital
artist has traced the outlines of the balloon from a photograph and then colored
the section of the balloon using vector based information.
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As you can see, when you zoom in on the balloon vector image, there is no
ugly pixilation – the image “behaves” will under any level of magnification.
It is was a raster image and we zoomed in, then the same magnification
would actually look quite ugly, like this
VECTOR IN PHYSICS
Euclidean Vector
In Mathematics, physics and engineering, a Education vector is a
geometric object that has magnitude and direction. Vector can be added to other
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vectors according to vector algebra. A Euclidean vector is frequently
represented by the line segment with the definite direction, or graphically as an
A vector is what is needed to “carry” the point A to the point Bi the latin
word vector means “carrier”. It was first used by 18 th century astronomers
investigating planet rotation around the sun. The magnitude of the vector is the
distance between the two points and the direction refers to the direction of
displacement from A to B. Many algebraic operations on real numbers such as
addition subtraction, multiplication, and negative have dose analogues for
vectors, operations which obey the families algebraic laws of commutativity
associativity and distributivity. These operations and associated laws qualify
Euclidean vectors as an example of the more generalized concept of vector
defined simply as elements of a vector space.
VECTORS IN MATHEMATICS
Vector Space
A vector space is a collection of objects called vectors, which may be added
together and multiplied by numbers, called scalars. Scalars are often taken to be
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real numbers, but there are also vector spaces with scalar multiplication by
complex number, rational numbers or generally any field, the operations of
vector addition and scalar multiplication must satisfy certain requirements
called axioms.
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upward could be represented by the vector (0,5). Another quantity represented
by a vector is force, since it has a magnitude and direction and follows the rules
of vector addition vectors also describe many other physical quantities, such as
linear displacement, displacement, linear acceleration, angylar accerlation,
linear momentum, and angular momentum. Other physical vectors, such as the
electric and magnetic field are represented as a system of vectors at each space,
that is, a vector field. Examples of quantities that have magnitude and direction
but fail to follow the rules of vector addition : Angular displacement and
electric current.
APPLICATION OF VECTORS
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Cannon
A cannon is any piece of artillery that uses gun powder or other usually
explosive based propellan to launch a projectile – cannon vary in calber, range,
mobility, rate of fire, angle of fire, and the fire power; different forms of anon
combine and balance these attributes in varying degrees, depending on their
intended use on the battlefield. Of course for this we need vectors. It is used
also in aircraft. The first documented installation of a cannon on an aircraft was
on the voisin cannon in 1911, displaced at the paris exposition that year. By
world war I, all of the major powers were experimenting with aircraft mounted
cannons; however their low rate of fire and great size and weight preduded any
of them from being anything other than experimental.
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In football match, player who want to score a goal, he cant shoot ball 10
meters left, and after that 9 meters right, it is impossible, so here we need
vectors to determine the direction or trajectory of ball.
Also in basketball match, this is the same. For throwing a ball through a
nettled hoop, again you have to know the direction or trajectory of ball.
Golf is also same, but according to golf ball is small, you must consider
the winds the vector of wind force. It wind is strong you must consider the
wind force also in football.
Wind Vectors
Leys, we have plane with constant velocity, and plane move to south and
we have wind force which direction of it is west, so due to plane movement is
south and wind movement is west. Finally plane move diagonally, or in the
south west.
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the fact that you are moving through the air. This is the apparent wind. On the
windless day, the apparent wind will always be directly in front and equal in
speed to the speed of the bicycle.
In sailing, the apparent wind is the actual flow of air acting upon a sail.
It is the wind as it appears to the sailor on a moving vessel. It differs in speed
and direction from the true wind that is experience by a stationary observer. In
nautical terminology, these properties of the apparent wind are normally
expressed in knots and degrees. On boats, apparent wind is measured or “felt
on face/ skin” if an dinghy or looking at any telltales or wind indicators on
board. True wind needs to be calculated or stop the boat.
Windsurfers and certain types of boats are able to sail faster than the true
wind. These include fast multinulls and some planning monohulls. Ice sailors
and land sailors also usually fall into this category, because of their relatively
low amount of drag or friction.
Force, Torque, Accerlation, Velocity and etc.
For calculating every vectorial unit, you need vector. For example, there
is a tire with mass m, and it has initial and final velocity, acceleration,
gravitational, reaction, friction forces, and due to rotation it has torque. For
getting the result, you need vectors. May be it seems like boring problem, but
we need it in daily life, for instance, finding velocity of acceleration of cars. In
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construction, every architect have to know their buildings of durability, for this
they need forces that max how many forces will apply to their building, and of
course they need again vectors. So we can see how the vectors are important.
Roller Coaster
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By doing this project on the topic application of vectors, I could find out
more and wide range of application of the topic in various fields which were
unknown for me before.
Vectors have applications in various areas from daily life to science and
technology. Most of the people are unaware of its importance or not making
use of its due to the lack of knowledge in the topic. As this project helped me in
widening my knowledge regarding the topic, I could came to the conclusion
that it also help the students in improving their knowledge and help them apply
for the progressive development of the society.
BIBLIOGRAPHY
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Class XI - CBSE
2. Mathematics text book
Class XII – SBSE
Websites
Wikipedia
www.quora.com
www.intmath.com
www.google.co.in
www.mathinsight.org
APPENDIX
Pencil
Pen
Scale
Pictures
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Paper cuttings
Rubber
Sketch pens
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VECTORS AND ITS REAL LIFE
APPLICATIONS
INDEX
1. Abstract 1
2. Introduction 2
3. Objectives 4
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4. Hypothesis 5
6. Applications of Vector 13
8. Bibliography 26
9. Appendix 27
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