kps vector

7/31/2019 Kps Vector

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VECTORS


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Why do we need vectors?

During the process of understanding andanalysing laws of nature physicists havedeviced several interesting and usefull

physical quantities. These quantities are usedto describe natural phenomena scientifically

which has developed as a universal language.Nature communicates with man using this

language. Proper understanding of thislanguage is indispensible for any student of

science.


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Some physical quantities can becompletely described by specifying

their magnitude in proper units. Someothers require specification ofmagnitude as well as direction for

their complete description. Theformer are called Scalars and thelatter are called Vectors.

Why do we need vectors?


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Examples of Scalars

If you meaure thetemperature of your room

using a thermometer youexpress it as a number

having a unit like celsious ,Farenheit etc. These

numbers can be combinedusing ordinary laws of

arithmatic.


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Examples of Scalars

If you meaure the weight ofpotatos using a Kichen

Balance you express it as anumber having a unit like

kilogram , pound etc. Thesenumbers can be combined

using ordinary laws ofarithmatic.


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Examples of Scalars

If you meaure your wrist

size using a tape measureyou express it as a number

having a unit likecentimetre, inch etc. Thesenumbers can be combined

using ordinary laws ofarithmatic.


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Commonly used Scalars

Quantity Unit Quantity Unit

Mass Kilogram Energy Joule

Distancetravelled

Metre Power Watt

SpeedMetre/second

ElectricPotential

Volt

Timeinterval

Second Temperature Celsius


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In a football game the ball is passed from theinitial position A to final position D as shown.The actual distance the ball has travelled is

the distance AB + distance BC + distance CDwhich is a number and hence a scalar

expressed in length units. But thedisplacement is the distance measured along

the line AD. It has a magnitude (distance AD),a length unit, and a direction (along the line

AD). Hence it is a vector.

Examples of Vectors


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A

B

D

C


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Mathematics to help you...

Mathematical idea behind vector manipulations iscalled vector analysis. In order to develop

mathematical methods to handle vectors we usedisplacement vector as a handy prototype.

Methods so developed can be conveniently usedin handling other vectors like velocity, force,

electric field etc. without any change in spite of thefact that they are different quantities.Maths is powerful...


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Commonly used Vectors

Quantity Unit Quantity Unit

Force newton Acceleration m/s2

Displacement Metre Electric dipolemoment

Coul-m

VelocityMetre/second

Magneticfield Tesla

Electricfield vector

Volt/Metre PositionVector

Metre


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How will you represent a Vector?

Simplest representation of a vector is a directedline segment (arrow) with the length of the

segment representing the magnitude of the vector(in suitable scale) and the direction of the arrow

head pointing towards the direction of the vector.For example force acting on a 10 kg. Mass on the

surface of the earth (g=10m/s

2

, scale 10N=1cm.)can be represented by a vertically downwardarrow of 10cm. length.


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When are two vectors equal?

Two vectors are said to be equal only if theyhave same magnitude and direction. Thus

when we say a vector changes it means that

magnitude, direction or both changes. Ingraphical representation two equal vectors are

represented by two parrallel arrows of samelength.

If two vectors have equal magnitude andopposite direction one is said to be the

negative of the other (A and -A).


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Graphical representation

Vector A

Vector B

Vector C

D = -C

O

T

Vector P

Vector A and Vector B are equal inmagnitude and direction hence theyare equalVector D and Vector C are equal andopposite hence D is the negative of

CVector P is represented by the arrowOT. The point O is called the initialpoint or origin and the point T iscalled the terminal point orTerminus.The distance between initial point

and terminal point represents themagnitude of the vector (the lengthbeing taken after pre-fixing aconvenient scale.)


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Algebraic operations on vectors...

As in ordinary algebra operations like Addition,Subtraction, multiplication with a number,

multiplication among themselves etc. are defined.They form the rules of vector algebra.


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Adding two parallel vectors

If you add two parallel vectors ie. two vectors in

same direction the resulting vector is anothervector in the same direction having magnitude

equal to the sum of the magnitudes.Obviously the rule can be extented to any number

of parallel vectors.


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Let us do it...

Velocity Vector A 6m/s

Velocity Vector B 4m/s

|A| = 6m/s|B| = 4m/s|A+B| = 10m/s

|A| indicatesmagnitudeof A

Velocity Vector A 6m/sVelocity Vector A 6m/sVelocity Vector A 6m/sVelocity Vector A 6m/sVelocity Vector A 6m/s

Velocity Vector A 6m/s + Velocity Vector B 4m/s

C=A+B


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Let us do it...

|P|= 3|Q|= 4|R|= 6|P+Q+R| = 13

P = 3

Q = 4

R = 6

P + Q + R = S

S


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Resultant of Vectors...

Two or more vectors [system of vectors] whenapplied produces a cobined effect. A single vector

which produces the same effect as that of thesystem of vectors is called resultant of the system

of vectors. To find the resultant we add all the

members of the system of vectors according tothe rules of vector algebra.


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Adding equal vectors

Thus multiplication of a vector by an integer n giving nA isequivalent to addition of n equal vectors. This idea can be

generalised to mA where m is any scalar positive ornegative. This is called multiplication of a vector by a

scalar. Multiplication by positive scalars do not change thedirection. Magnitude gets multiplied by the scalar.

Multiplication by negative numbers give vectors in theopposite direction with magnitude multiplied by the scalar.

A

A

A

A+A+A = 3A


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Subtracting two parallel vectors

If you subtract one parallel vector from another

the resulting vector is another vector in the samedirection as that of the larger vector having

magnitude equal to the difference of themagnitudes.


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Let us do it...

|A| = 6 N|B| = 4 N|A-B| = 2 N

Force Vector A 6 N

Force Vector B 4 N

C = A - BThe direction of the resultant inthis case will be the direction ofVector having larger magnitude

2m/s

C = A - B

10 N 15 N

5 N


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Dimension of a Vector...

If all the vectors in the system are parallel wehave a system of vectors in one dimension.

If all the vectors in the system are not parallel

and if they all lie in the same plane [coplanarvectors] we have a system of vectors in twodimension.

A system of vectors in which one ore more ofthe vectors do not lie in the plane it is called athree dimensional system of vectors.

if di i i


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How to specify direction usingangle with the horizontal ?

So far we were dealing with parallel vectors. Theyare all one dimensional cases where the direction

can be specified by a sign either + or after fixinga suitable convention. Most popular convention isto assign + to vectors towards right or up and tovectors towards left or down. But vectors in two or

three dimension [ vectors which are not parallel]demands a more mathematical approach.

H if di i i


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How to specify direction usingangle with the horizontal ?...

Horizontal direction (hd)

Verticaldir

ection

(vd)

Direction of the vector can bespecified by taking the anglemade by the vector with thehorizontal in the

anticlockwise direction.


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How to determine magnitude ?...

Horizontal direction

Verticaldir

ection

Magnitude of the vector canbe determined by measuringthe horizontal distance andthe vertical distance [both are

scalars] and usingPythagoras theorem.

Horizontal distance

Verticaldistance


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Addition of vectors Triangle law

Displace one of the vectors (second vector)such that the tail (origin or initial point) of itcoincide with the head (terminus or final point)of the other (first vector).

Draw a vector connecting the tail of the firstvector and head of the second vector.

This vector represents the sum both in

magnitude and direction.

kps vector

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