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SCALARS AND VECTORS SCALARS AND VECTORS Vikasana – Bridge Course 2012 1

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Page 1: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

SCALARS AND VECTORS SCALARS AND VECTORS

Vikasana – Bridge Course 2012 1

Page 2: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

Quantity : Any thing that is measurable is termed as ‘quantity’termed as quantity

Physical Quantity : The quantities that comeacross in physics is referred as a PhysicalQuantity.

E M l th t t ti tEx : Mass, length, temperature, time, etc.,

Whenever we measure a physical quantity theWhenever we measure a physical quantity the measured value is always a number. This

number makes sense only when the relevant

Vikasana – Bridge Course 2012

yunit is specified.

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Page 3: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

Thus the result of measurement has aThus, the result of measurement has anumerical value and unit measure.

Ex : The mass of the body 3 kgHere a quantity have a numerical value 3Here a quantity have a numerical value 3And unit of measure kg are usedThe numerical value together with the unitThe numerical value together with the unit

called the ‘magnitude’

Vikasana – Bridge Course 2012 3

Page 4: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

To describe certain physical quantities like 

mass, distance, time, temperature, no books 

in a bag are some examples of numbers. they 

h di ti i t ith thhave no direction associate with them. 

Based on Magnitude and direction physicalBased on Magnitude and direction physical 

quantity can be classified to two types.   

Vikasana – Bridge Course 2012

q y yp

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Page 5: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

h i l i iScalars

Physical QuantitiesVectors

A scalar is completely described by a number.

E g mass (m) temperature (T) etcE.g. mass (m), temperature (T), etc

A vector is completely described by :

Its magnitude Its direction

Vikasana – Bridge Course 2012E.g. A car moves with a speed 400 km/h towards Sira

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Page 6: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

Representation of a vector : •A Vector can be represented by a straight line with arrow h dA

a

head.•The length of the vector represents its magnituderepresents its magnitude and arrow indicates its direction

o •The vector represented by the line segment OA and it

Vikasana – Bridge Course 2012is denoted by OA.

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Page 7: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

A Simple notation as (to be read as vector a).

For vector OA O is the initial point and A is the

a

For vector OA , O is the initial point and A is the

terminal point.

The magnitude of OA or denoted by |OA|,

( d d l f t OA) d i l( read as modulus of vector OA) and is always

positive

Vikasana – Bridge Course 2012 7

Page 8: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

Position Vector To locate the position of anPosition Vector ; To locate the position of any point ‘p’ in plane or space, generally a fixed point of a reference called the origion ‘O’ ispoint of a reference called the origion O is taken.

P(x,y)Y

r

The vector OP is called the position vector of p with

t t O h i

O X

respect to O as shown in the fig. 2

Fig - 2

Vikasana – Bridge Course 2012

O XFig - 2

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Page 9: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

Kinds of VectorsKinds of Vectors 1) Unit Vector : A vector having unit

magnitude is called unit vector

If A is a vector having magnitude |A| ≠ 0, then g g | | ,

A is a unit vector having a same direction |A| as A It is represented as A ( read as cap)^|A| as A . It is represented as A ( read as cap)

|A | = 1 , then A = A or |A | A = A A^ ^ ^ ^

Vikasana – Bridge Course 2012

| | , | ||A|

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Page 10: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

Note : The unit denoted as vectors along the x, y ^ ^ ^and z - axis are usually i , j and k respectively.

2) Zero vector or Null vector : A vector having Zero magnitude null vector or Zero vector. It is represented as o .

N tNote : a) Zero vector has no specific direction.b) Th iti t f i i i tb) The position vector of origin is zero vector.c) Zero vectors are only of mathematical

Vikasana – Bridge Course 2012

importance.10

Page 11: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

3) Equal vectors : Vectors are said to be equal if ) q qboth vectors have same magnitude and direction.

a

b

4) Parallel Vectors (Like Vectors): Vectors area = b

4) Parallel Vectors (Like Vectors): Vectors are said to be Parallel if they have the same directions. a

Vikasana – Bridge Course 2012b

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Page 12: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

The vectors a and b represent parallel vectors.

Note : Two equal vectors are always parallel but,two parallel vectors may or may not be equaltwo parallel vectors may or may not be equalvectors.The vectors a and b represent parallel vectors.p p5) Anti parallel vectors( unlike vectors) : Vectorsare said to be anti parallel if they act in oppositedirection.

a

Vikasana – Bridge Course 2012

bThe vectors a and b anti parallel vectors

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Page 13: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

6) Negative Vector : The negative vector of) g gany vector is a vector having equalmagnitude but acts in opposite direction

a

b

a = - b or b = - aa b or b a

Vikasana – Bridge Course 2012 13

Page 14: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

7) Concurrent Vectors (Co initial Vectors): ) ( )Vectors having the same initial points are called concurrent vectors or Co initial vectors.

A OA

o

A

BC

OA = a

OB = bo C OC = c

b d t t i t ‘ ’Vikasana – Bridge Course 2012

a , b and c are concurrent at point ‘o’14

Page 15: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

8) Coplanar Vectors : The Vectors lies in the ) psame plane are called coplanar vectors.

a

b

c

The Vectors a, b and c are called coplanar t

b

vectors.

Vikasana – Bridge Course 2012 15

Page 16: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

9) Orthogonal Vectors : Two Vectors are said to ) gbe Orthogonal to one another if the angle between them is 900

b

y

b

ao x

The vectors a and b Orthogonal to one another

a

Vikasana – Bridge Course 2012

The vectors a and b Orthogonal to one another

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Page 17: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

VECTOR ALGEBRA Addition of Vectors : The addition of scalarsinvolves only the addition of their magnitudes.But When vector is added with another vector weBut, When vector is added with another vector wehave to consider the direction also .

Vector can be added with another vectorVector can be added with another vectorprovided both the vectors represents the samephysical quantity.

Example the addition of a vector representingthe displacement of a body with another vectorrepresenting elocit of the bod is meaning

Vikasana – Bridge Course 2012

representing velocity of the body is meaningless.

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Page 18: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

Method of Vector Addition1. Triangle method of vector addition:Illustration : CIllustration :

a b =+c b

a + b = c

Add vector a with b, translate b, byA Ba

Add vector a with b, translate b, by drawing parallel to itself. So that the origin or initial point of b is at the tip of

Vikasana – Bridge Course 2012

origin or initial point of b is at the tip of vector a.

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Page 19: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

a and b are two vectors represented by two sides of a triangle taken in the ordertwo sides of a triangle taken in the order, then resultant of a and b is represented by the third side of the triangle is taken in thethe third side of the triangle is taken in the reverse order.Statement : if two vectors can beStatement : if two vectors can be represented in magnitude and direction by two sides of a triangle taken in the ordertwo sides of a triangle taken in the order, then their resultant is represented by the third side of a triangle taken in the

Vikasana – Bridge Course 2012

third side of a triangle taken in the opposite order

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Page 20: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

Parallelogram method of vector addition  

By law of triangle method of vector 

Vikasana – Bridge Course 2012addition R = A+B or R= B+A

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Page 21: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

Statement of parallelogram of vectoraddition : “If two vectors acting at apoint can be represented both inp pmagnitude and direction by the twoadjacent sides of a parallelogramadjacent sides of a parallelogramdrawn from that point, the resultant isrepresented completely by therepresented completely by thediagonal of the parallelogram passingth h th t i t”

Vikasana – Bridge Course 2012

through that point”.21

Page 22: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

It can be shown that magnitude of R is given by R= A2+B2+2ABCOSθ

If α is the angle made by the direction of R g ywith that of B then

A sin θA sin θB +A cosθ

N t ) C t ti l f t dditi

tanα =

Note : a) Commutative law of vector additionA+B = B+A

Vikasana – Bridge Course 2012b) Associative law of vectors: a + (b+c) =

(a+b)+c 22

Page 23: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

Law of polygon of Vectors AdditionStatement : “ States that if a number of

vectors are represented both inpmagnitude and direction by the sides ofa polygon taken in the order, then theirp yg ,sum ( resultant) is represented both inmagnitude and direction by the closingg y gside of the polygon taken in theopposite order”

Vikasana – Bridge Course 2012

pp

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Page 24: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

Let vectors a b c and dC

Let vectors a, b, c and d

represents OA, AB, BC and

BD

R

d c

CD of the polygon OA B C D

O A

R b

aThen OD = OA + AB+ BC +CDThen OD = OA + AB+ BC +CD

Vikasana – Bridge Course 2012 24

Page 25: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

Subtraction of two vectorsSubtraction of one vector b from another

vector a by using negative of vectors y g gas follows

+a

- =a

-b=

b

Vikasana – Bridge Course 2012 25

Page 26: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

By triangle method

aa b

b-

- =b

a b- a

Note : The knowledge of subtraction of vectors is useful in understanding the concept of “relative velocity”

Vikasana – Bridge Course 2012 26

Page 27: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

RESOLUTION OF VECTORS The process of splitting of a vector intoits components. “If a vector is resolvedpinto two components along the twomutually perpendicular direction arey p pcalled rectangular components”.

YCB C

RQ

θ

Vikasana – Bridge Course 2012

OAP

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Page 28: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

Let the vector R makes an angle θ with X i D di l f th tiaxis. Draw a perpendicular from the tip

of vector C to X and Y axes OA =P is the t R l i d i ll dcomponent R along x axis and is called

horizontal component of R.OB = Q is the component of R along Y

axis and is called vertical component of ^ ^R. then R = P+Q = i P + jQ

Where i and j are unit vectors act in along

^ ^^^

Vikasana – Bridge Course 2012

j gx and y axis.

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Page 29: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

The magnitude of OA is

∴ P = R COS θ

OAOC = COSθ ⇒ OA = OC COS θ

∴ P = R COS θThe magnitude of OB is

AC

∴ Q = R Sin θ

ACOC = Sin θ ⇒ AC = OC Sin θ

∴ Q = R Sin θAlso AC

OA =tanθ ⇒ tan θ = QP

Vikasana – Bridge Course 2012

OA P29

Page 30: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

From geometry of the figure, it can be h th t R 2 P2 Q2shown that R 2 = P2+Q2

**************************************************

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Page 31: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

Assignment Questions 1. What is a scalar?2. Identify whether the following quantities y g qare scalars or vectors ?

(1) Mass (2) weight (3) Speed( ) ( ) g ( ) p(4) velocity (5) energy (6) Work(7)temperature (8) pressure (9) Force( ) p ( ) p ( )

3. What is vector ? 4. Find the magnitude of 4 i + 3j ^ ^

Vikasana – Bridge Course 2012

g j

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Page 32: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

5. Two vectors P and Q act at angle 600Q gwith each other. If P = 20 units and Q = 8 units find the magnitude of resultant R g6. If A = 6 i + 3j and B = 3i + 2j find A +B and A-B

^^

^ ^ ^ ^

7. If A = 3i -2j +k, find unit vector C8. When will be P + Q= 0

^^^

Q

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Page 33: SCALARS AND VECTORS · The vectors a and b represent parallel vectors. Note : Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors. The

9. A vector of magnitude 10 units makes

an angle of 300 with X- axis. Find its X and

Y components.

10 For what angle of magnitude of X and10. For what angle of magnitude of X and

Y components of vector becomes equal?

Vikasana – Bridge Course 2012***************

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