1. Vector Algebra1

7/27/2019 1. Vector Algebra1

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VECTOR ALGEBRA


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Some quantities that you have studiedin your earlier Physics Courses

volumemass

density

energypressure

displacementvelocity

acceleration

torqueelectric field


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Scalar Quantities

50 kg 40 C

A scalar is a number which defines

a magnitude.

It does not point to any direction in space.

Examples: mass, volume, speed, density,temperature, energy, electrical potential

and charge.


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What are vectors ?

A physical quantity which has both a

magnitude and a direction is represented by a

vector.

Examples: moment of inertia, acceleration,

force, velocity, displacement, momentum,

magnetic induction, magnetic and electric

intensities.

i. A vector has a both a magni tudeand a

direct ion.

ii. It must satisfy the parallelog ram law of

addit ion.


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A

B

Displacement Electric Field

+

Examples


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Vector Notation

x, F, v, a

Vectors are written as a symbol

with an arrow over the symbol

Magnitude of a vector quantitycan be written as: a


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Representation

The length of the arrowrepresents the magnitude ofthe vector quantity.

A vector is represented by anarrow.

The arrow points in thedirection of the vector.


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5 mA

6 mA

2 mA

5 mA

6 mA

11 mA


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Addition of Vectors

Triangle Law

a

b

c

a

bc = a + b

Head-to-tail


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a

b

c

Commutative Law

Addition of Vectors

cc = b + a


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Addition of Vectors (contd.)

aa

bb

c

a

bc

Parallelogram Lawc = a + b


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ab

da

b

cd

ec

ef

f = a + b + c + d +e

Head-to-tail


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a

d

c

b

a

c

b

cbae

)(

e


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Associative Law

e

a

c

b

b

c

a

d

e

)( cbae


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Subtraction of Vectors

ab

a-b

c

c = ab = a +(b)


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ADDITIONOF VECTORS (CONTD.)

Can be manipulated as in

usual algebra

c = a

b

a = b + c


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Vectors are geometrical objects

independent of any

coordinate system

Let us look for a convenient descr ipt ion !


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2D Cartesian Coordinates

O

X

Y

a

ax

ayy1

y2

x1 x2

Look a two dimensional vector in a

2D Cartesian Coordinate System

d ( d )


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Y

OX

a

ax

ay

a = ax2 + ay2ax=x2x1= acosay=y2y1= acos

2D Cartesian Coordinates (contd.)

ay

ax

2D d ( d )


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Both the magnitude

and the direction of a arecompletely specif ied byax& ay

Could I then say that the set of

numbers (ax, ay) is a complete

description of the vector a?


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OX

Y

a

ax

ay Unit vectors: i , j

Components: ax ,

ay

a = axi +ayj


Direction cosines: cos, cos


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More on Direction cosines

The direction cosines of a vector arenot independent. They satisfy the

following relation:

1

coscos

2

2

2

2

22

aa

aa yx


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VECTOR COMPONENTS

Coordinate System

Resolving a vector :The process of finding the

components of the vector.


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The component of the vectoralong an axis is

its projection along that axis

Component of a vector

O x

ya

ax

ay


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3D Cartesian System

ax

ay

azj

i

k

A

O

X

Y

Z


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3D Cartesian System (contd.)

a = axi +ayj +azk

a = ax2 + ay2+ az2ax=x2x1= acosay

=y2

y1

= acosaz=z2z1= acos

coscoscos

Direction

Co

sines


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3D Cartesian System(contd.)

The direction cosines satisfy thefollowing relation:

1

coscoscos

2

2

2

2

2

2

222

a

a

a

a

a

a zyx


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Unit Vectors

Indicates directiononly

a = i +6j + k

38

a = i +6j + k

a = a a

Magnitude = 1

a =12 + 62 +12 = 38

a = a /a


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aba =axi +ayj +azkb =bxi +byj +bzk

c = a + b

=(ax+bx)i +(ay+ by)j

+(az+bz)k

=cxi +cyj +czkcx=(ax+ bx) cy= (ay+ by)

cz= (az+ bz)


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Examples

a =4i +3j +2k

b =3i -2j + kc =7 i + j +3kc = a + b

c = a - b c = i +5j + k


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O

R

0.

8Km

1.5

Km0.2 Km

d

d = 0.8 j + 0.2 i + 1.5 j= 0.2 i + 2.3 j

Examples

x

y


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SUMMARY

A physical quantity which has both amagnitude and a direction is represented

by a vector

A geometrical representation

An analytical description: components

Can be resolved into components along

any three directions which are non

planar.


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Vector operations:

1. Addition and subtraction of two vectors:

Addition:By placing tail of B at the head ofA, the sum vector (R), A+B is drawn from

the tail of A to the head of B

AB

R

R A B


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Subtraction:By placing tai l of B at the

head of A, the resultant vector (R), A-B isdrawn f rom the tail of B to the head of A

Addition is commutative:

Already discussed

R A B

A B B A

A

B

R


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2. Mul tiplication by a scalar : Scalar

mul tipl ication is distr ibutive.

I f a is negative, the direction is reversed.

3. Scalar Product or Dot product of two

vectors:is a

scalar quanti ty.

where is the angle they formwhen placedtail to tai l.

. cosA B AB

( )a A B aA aB


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Dot product is also distr ibutive:

Physical Signi f icance of :

is the product of A times the projection of

along or the product of B times the

projection of along.

.( ) . .A B C A B A C

.A B

B

A

A

B

Dot product is Commutative: . .A B B A


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Let two vectors are parallel:

Let two vectors are perpendicular:

4. Cross product of two vectors is itself avector:

here n is a unit vector pointingperpendicular to the plane of and

.A B

.A B

sinA B AB n

A B

AB

0

Th di ti f b f d b i ht


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The cross product is distr ibutive:

The cross product is not commutative:

Cross product gives the area of the

parallelogramgenerated by and

The direction of n can be found by r ight

hand rule:If fingers point in the direction of

the first vector and curl around toward the

second, the thumb gives the direction of n.

( ) ( ) ( )A B C A B A C

( ) ( )A B B A

A B

V t l b C t f


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Vector algebra: Component form

Let and be unit vectors parallel to the

x, y, and z axes respectively.

Any vector A can be expanded in terms of

these basic vectors, with Ax, Ay and Azare

components of A:

Here, Ax, Ayand Azare the projections of A

along the three coordinate axes.

x y zA A x A y A z

,x yz

R l 1 T dd dd lik


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Rule 1: To add vectors, add like

components.

Rule 2: To multiply by a scalar, multiply

each component.

( ) ( )x y z x y zA B A x A y A z B x B y B z

( ) ( ) ( )x x y y z zA B x A B y A B z

( ) ( ) ( )x y z

A aA x aA y aA z


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Rule 3:To calculate the dot product, multiply

like components, and add.

Rule 4:To calculate the cross product, formthe determinant whose first row is ,whose

second row is A (in component form), and

h hi d i

. ( ).( )x y z x y zA B A x A y A z B x B y B z

x x y y z zA B A B A B

. . . 1

. . . 0x x y y z z

x y x y x z

Because and are mutuallyperpendicular unit vectors.

,x yz

, ,x y z

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