phys201 11314 chapter1 vectoralgebra -...

http://ctaps.yu.edu.jo/physics/Courses/Phys201/

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Methods of Theoretical Physics 1

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Chapter 1

Vector Analysis

http://ctaps.yu.edu.jo/physics/Courses/Phys201/Chapter1

Vector Algebra Scalars and Vectors

© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra

5

Scalars – Scalar Fields

A Scalar is a physical quantity completely

defined by its magnitude.

Examples are: mass, temperature, pressure…

A scalar field is a scalar mathematical function

that defines a certain physical quantity at

every point in a given region of space.


6

A Vector is a mathematical representation of a

physical quantity which should be defined by

its magnitude and direction

Vectors – Vector Fields

Examples are: Force, linear momentum

A vector field is a vector mathematical function

that defines a physical vector at every point in

a given region in space.


7

→A

A vector is represented by an arrow. Its length is

proportional to the length of the quantity it represents.

The arrow points to its direction.

Graphical representation of a vector 1

The vector makes an angle of 45°°°° with the

direction representing the east (E)

→A

The vector represents

the wind speed of

60 km/h whose direction

is North-East.

1 unit = 10 km/h

→A

6 un

its

N

EW

S

The symbol is used to represent a vector.→A


8

A Vector has no position. Thus the vector

also represents the wind speed of 60 km/h whose

direction is North-East.

→B

Graphical representation of a vector 2

→A

6 u

nits

6 u

nits

→B

N

EW

S


9

Length of a vector

We represent the length of a vector by:

| | or simply A.

In the previous example | | = 60 km /h→A

→A

6 u

nits

In this course we shall indifferently use | | or A

to represent the length of a vector

→A

→A

→A

Properties of Vectors


11

The Vector Space

Mathematicians have devised some operations

involving vectors that pertain to physical quantities

that have both magnitude and direction

Mathematicians define a space called the vector

space which has its own “rules” and operations.

As for any space we shall define in the vector space

some properties and mathematical operations such

as addition, subtraction, multiplication, derivation,

etc…

See Supplement 1


12

Properties of vectors1

a) | | = | |, i.e. and have the same magnitude.→A

If and only if:→→

= BA1)

→B

→A

→B

b) // , i.e. and have the same direction.→A

→B

→A

→B

N

EW

S

→A

6 u

nits

6 u

nits

→B


13

Properties of vectors2→→

−= BA If and only if:2)

b) anti// , i.e. and have opposite

directions.

→A

→B

→A

→B

N

EW

S

→A

6 u

nits

6 u

nits

→B

a) | | = | |, i.e. and have the same magnitude.→A

→B

→A

→B

Addition of Vectors


15

We say that vector is the sum (or resultant)

of two vectors and by writing:

Addition of vectors

Two (equivalent) methods are used to add two

vectors:

• The head to tail (or the triangle) method.

• The parallelogram method.

+→A

→B

→C =

→A

→B

→C


16

In this method we draw, starting from the head of

vector (point a) a vector equal to . (b is the head

of the latter).

The head to tail (or the triangle) method

→A

→B

→A

→B

→C

→B

→A

O

a

b

The sum (vector ) is the vector

completing the triangle with b as its head.

→C

→Ob


17

In this method we draw, starting from the tail of

vector (point O) a vector equal to . (b is the

head of the latter). The sum (or vector ) is the

diagonal of the parallelogram formed by the 2

vectors.

The parallelogram method

→A

→A

→B

→C

→B

→Oc

→C

→A

O

a

c

→B

b

→B

→A

Subtraction of Vectors


19

To subtract vector from vector , we use

the definition of

→B

Subtraction of vectors

The previous methods are used to obtain

→A

)B(ABAC→→→→→

−+=−=

→−B

→C


20

The head to tail (or the triangle) method

(or vector ) is the

vector completing the

triangle with b as its

head.

→C

→Ob

In this method we draw,

starting from the head of

vector (point a) a vector

equal to .

→A

→B

O

→A a

→B-

→A

→−B

b


21

In this method we

draw, starting from the

rear of vector (point

O) a vector equal to - .

The parallelogram method

→A

→A

→B

→B

O

→A

→A

a

b

→B

-

c

(or vector ) is the

diagonal of the

parallelogram formed by

the 2 vectors and - .

→C

→Oc

→A

→B

→C

Representation of Vectors

in a Cartesian System of

Coordinates


23

In a right-handed Cartesian system the axes Ox, Oy

and Oz are mutually perpendicular and they verify

the right-hand rule in this order. (conventionally

counter-clockwise)

Right-handed Cartesian System

x

z

y

Back to Components© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors - Vector Algenra

24

Representing vectors is independent of the

coordinate system we use.

Components of a vector

→A

Projections of a

vector on the axes

of a coordinate

system are called

componentscomponents

x

z

y


25


→A

Ay

Az

Ax


26

αααα, ββββ and γγγγ are called the angles of direction.

Direction cosines of a vector

→A

α

β

γ

x

z

y

cosαααα, cosββββ and cosγγγγ are called the direction cosines.

Az

Ax

A

AA xx ==αcos| |→A

A

AA yy ==βcos| |→A

A

AA zz ==γcos| |→A

Ay


27

In a right-handed Cartesian system the

components of vector are denoted Ax , Ay and

Az.

Direction Cosines and Components

→A

α

β

γ

x

z

y

Ax = A cos αααα

Ay = A cos ββββ

Az = A cos γγγγ

→A

α = (Ox, )→A

γ = (Oz, )→A

β = (Oy, )→A

Ay

Az

Ax


28

A and the components

→A

Ay

Az

Ax

22

yx AA ++++

O

P

Q

z

y

x

Triangles ORQ, OTP and

OPQ are right angled.

T

R

OR = PQ = Az

22

yx AA ++++RQ = OP =


29

A and the componentsBy Pythagorean theorem, we have:

( ) 21222zyx AAAA ++=

22

222

yx AA ++++====

++++==== TPOTOP

(((( )))) 2222

222

zyx AAAA ++++++++====

++++==== QPOPOQ

OR = PQ = Az

22

yx AA ++++RQ = OP =


30

The previous relations show that if we know the

three components Ax, Ay and Az then we know

exactly the vector itself.

Defining a vector using its components

→A

222zyx AAAA ++=

A

AA yy ==β→

|A|

cos

A

AA xx ==α→

|A|cos

A

AA zz ==γ→

|A|cos

Multiplication in the

Vector Space


32

We define 3 types of multiplication in the

vector space, namely:

Multiplication of a vector by a scalar

• Multiplication of a vector by a scalar.

• The scalar product: here the multiplication

of 2 vectors gives a scalar.

• The vector product: here the multiplication

of 2 vectors gives a (3rd) vector.

Scalar ×××× Vector


34

Multiplication of a vector by a scalar

Given a vector , we define, for any real

number m, the vector:

Obviously we have (m#0):

|||| AC→→

= ma)

→C

→A

m

1=

a) | | = | |→C

→A

m

1

→A

→→= AC m

b)If m > 0 then // ;

If m < 0 then anti //

→A

→C

→A

→C

b)If m > 0 then // ;

If m < 0 then anti //

→A

→C

→A

→C

= m A


35

For a particle of mass m moving with velocity ,

the linear momentum is defined by

a) | | = m | |

Example: Linear momentum

b) // , i.e. the 2 vectors have the same direction.

→→→→p

→→→→v

→p

→v

→p

→v

→→= vmp

→→→→v


36

A charge q when submitted to an electric field

suffers a force given by (Coulomb’s Law)

EqF��

=a)

Another Example: Electric force on a charge

EqF��

====

F�

E�

b) If q > 0 then // ;F�

E�

If q < 0 then anti // F�

E�

F�

E�

E�


37

One can use the previous property to define

the unit vector of any vector. Let’s consider

the vector defined by:

b)

a) a = 1

Unit Vectors

is called the unit vector in the direction of

vector

→a

→A

→→= A1aA

A

→→= Aa

→→A//a

→a

We use the following notation to write this unit vector.

Scalar (Dot) Product


39

We define the scalar product of 2 given vectors

as follows:

Vector times Vector = Scalar

The symbol •••• (dot) is used to indicate that the

product yields a scalar*.

Where symbolizes the angle between the

2 vectors.

( , )^→

A→B

* To distinguish it from the vector product which yields a vector as we said

=| | ×| | cos ( , )→A

→B

→A

→B•

→A

→B

^


40

Scalar product – Magnitude of a vector

We can easily see that

^

=| |2

→→→→=

→•

→A,AcosAAAA | |×| |

→A

The magnitude of vector is the square root of

the scalar product of by itself:

→A

→A

→→→•= AAAA = | |


41

=| | ×| | cos ( , )

The scalar product of 2 perpendicular vectors is

zero.

Orthogonal Vectors

→A

→B

→A

→B•

→A

→B

The 2 vectors are said to be orthogonal.

^

If = 90°°°° then = 0 ( , )^→

A→B

→A

→B•

→A

→B


42

- The scalar product is commutative; i.e.:

This is easily seen from the previous

definition. The reason is simply that

Properties of the scalar product

=→A

→B• •

→B

→A

cos ( , )→A

→B

^= cos ( , )

^→B

→A

- The scalar product is distributive; i.e.:

→→→→→→→•+•=

+• CABACBA

Vector (Cross) Product


44

We define the vector product of 2 given

vectors as follows:

| | =| | ×| | sin( , )→A

→B

^

, and form a right-handed system

Where symbolizes the angle between the 2

vectors.

( , )^

→A

→B

Vector × Vector = Vector

→A

→B

The symbol ×××× (cross) is used to indicate that

the product yields a vector. This product is also

called cross product.

××××××××→A

→B

→C= →

A→B

→C{

→C


46

A right handed system is defined (identified)

using the right-hand rule.

θθθθ

Right-handed Systems

→A

→B

→C


47

A B sinθ is nothing else but the area of the

parallelogram that the 2 vectors and form.

θθθθ

Area of a parallelogram – Area Vector

Thus vector is often called the area

vector of the previous parallelogram

→→→×= BAC

→A

→B

→B

→A

→C


48

≠→→

×AB→→

×BA

This is easily seen from the previous definition.

The reason is simply that

Properties of the vector product

- The vector product is not commutative; i.e.:

sin ( , )→A

→B

^≠ sin ( , )

^→B

→A

- The vector product is distributive; i.e.:

→→→→→→→×+×=

+× CABACBA


49

It can easily be seen that

= -××××××××→A

→B ××××××××

→B

→A

θθθθ

→B

→A

→→→→×−=× ABBA

sin ( , )→A

→B

^= - sin ( , )

^→B

→A


50

Scalar product - ExampleUnder the effect of an external electric field

an electric dipole rotates. The (rotational)

work done is given by:

Similarly under the effect of a magnetic field

a magnetic dipole rotates. The (rotational)

work done is given by:

→B

→→•= EpW

Where is the moment of this dipole.

→→•µ= BW

→E

→p

Where is the moment of the magnetic dipole.→µ


51

The rotation of the dipoles in the previous

examples is due to the torque (moment of the

force) defined by:

in the case of the magnetic dipole.

Vector Product - Example : Torque

in the case of the electric dipole and by:

→→→×=τ Ep

→→→

×µ=τ B

See Phys. 102

Cartesian Systems


53

In a cartesian system of coordinates, the

unit vectors in the directions x, y and z

respectively are called: , and . i j k

Cartesian Systems

z

yi j

k

x

© Dr. Nidal M. Ershaidat Phys. 201 Chapter 1: Vectors

54

Defining a Cartesian SystemAlternatively a Cartesian system is defined using

the 3 orthogonal unit vectors.

, and verify the following relations:i j k

z

yi j

k

kji ˆˆˆ =×

ikj ˆˆˆ =×

jik ˆˆˆ =×

0ˆˆˆˆˆˆ =•=•=• ikkjji

1ˆˆˆˆˆˆ =•=•=• kkjjii

These relations define the

orthogonality of the 3 unit

vectors’ system

II

x

Orthonormality


56

2 perpendicular unit vectors are said to be

orthonormalized (orthogonal and normalized)

Orthonormality

k

ji

0ˆˆˆˆˆˆ =•=•=• ikkjji

1ˆˆˆˆˆˆ =•=•=• kkjjii

These relations define the orthonormality of

the 3 unit vectors’ system


57

In a Cartesian system of coordinates, a vector

can be written as follows:


kji zyxÂÂÂA ++=

→

→A

γ

z

y

Ay

→A

appear as the sum of 3

vectors:

→A

Az

Ax

→→→→++= zyx AAAA

ixxÂA =

→

jyyÂA =

→

kzzÂA =

→ i j

k

x


58

→

zA

→→→→→→→→++++ yx AA

→

zA

→→→→++= zyx AAAA

→A

O

z

y

x

→

xA

→

yA

Scalar Product Using

Components


60

kji zyxˆBˆBˆBB ++=

→

Consider the scalar product of two vectors:


→

( ) ( ) ( ) 222 ˆBAˆBAˆBABA kji zzyyxx ++=•→→

zzyyxx BABABABA ++=•→→

222x

222x BBBAAA

BABABABAcos

zyzy

zzyyxx

++×++

++=

•=θ

→→

BA

If θθθθ is the angle between the 2 vectors then:

zzyyxx BABABABA ++=•→→


61

Consider the two vectors:

Example 2

ji ˆ3ˆ2A +=→

ji ˆˆ2B +−=→

( ) 11322BA −=×+−×=•→→

°≈−

=

+×+

−=

•=θ

−

−

→→

−

13.9765

1cos

1494

1cos

BAcos

1

11

BA



62

θθθθ

→A

→B

x

y

j

i

°≈θ 13.97


63

Consider the two vectors:

Example 3

kji ˆˆ3ˆ2A −+=→

kji ˆˆˆ2B ++−=→

( ) ( ) 2111322BA −=×−+×+−×=•→→

°=−

=

++×++

−=

•=θ

−

−

→→

−

6.10284

2cos

114194

2cos

BAcos

1

11

BA



64

kji zyxˆBˆBˆBB ++=

→

Consider the vector product of two vectors:

Vector product using components


→

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) kkjkik

kjjjij

kijiii

zzyzxz

zyyyxy

zxyxxx

ˆˆBAˆˆBAˆˆBA

ˆˆBAˆˆBAˆˆBA

ˆˆBAˆˆBAˆˆBABA

×+×+×

+×+×+×

+×+×+×=×→→

( ) ( )( )( )( ) ( )( ) ( )( ) 0ˆBAˆBA

ˆBA0ˆBA

ˆBAˆBA0BA

+−+

+++−

+−++=×→→

ij

ik

jk

yzxz

zyxy

zxyx


65

And rearranging we have:

( )( )( )k

j

i

xyyx

zxxz

yzzy

ˆBABA

ˆBABA

ˆBABABA

−

+−

+−=×→→


66

zyx

zyx

kji

BBB

AAA

ˆˆˆ

C =→

The previous result can be obtained using the

definition of determinants of matrices:

Vector product Using Determinants

kjiyx

yx

zx

zx

zy

zy ˆBB

AAˆ

BB

AAˆ

BB

AAC +−=→

Triple Products


68

Definition:

Triple Scalar Product – Mixed Product

→→→×•= CBAd

×•=→→→CBAd

This definition means precisely that:

But the parenthesis are not necessary since

the cross product is not defined→

×Cm


69

Properties of the Mixed Product→→→

×•= CBAd

zyx

zyx

zyx

CCC

BBB

AAA

=

( )( )( )( )

−+−+−

•++=k

j

i

kji

xyyx

zxxz

yzzy

zyx

ˆCBCB

ˆCBCB

ˆCBCBÂÂÂ


70

1.

We leave the proof of the following properties

as a homework.

- Cyclic permutation relation

→→→×• CBA

zyx

zyx

zyx

CCC

BBB

AAA

→→→×•= BAC

→→→×•= ACB

zyx

zyx

zyx

BBB

AAA

CCC

=

zyx

zyx

zyx

AAA

CCC

BBB

=

2. The following three determinants are strictly equivalent

→→→×• CBA

→→→→→→×•−=×•−= CABBCA

3. Any modification of the order of the 3 vectors

yields a negative sign, i.e.:

HW

2


71

θθθθO c

b d

φφφφ→A

- The mixed product represents in fact

the volume of the parallelepiped formed by the

3 vectors.

- Volume of a parallelepiped→→→

×• CBA

Vparallelepiped = B C sinθθθθ A cosφφφφ.

B C sinθθθθ = Area of the base of the

parallelogram (Obdc)

→→×CB

→B

→C

A cosφφφφ = Height of the parallelepiped


72

Definition:

Triple Vector Product – The BAC-CAB Rule

××=→→→→CBAD

It can easily be seen that:

We leave, as an exercise, the proof of this

relation known as the BAC BAC –– CAB ruleCAB rule.

××=→→→→CBAD

•−

•=

→→→→→→BACCAB

We can also use the properties of the vector

product to predict the result of the triple

vector product.

HW

2


73

Geometrical Justification of the BAC-CAB Rule

And is a vector which should lie in

the previous plane and thus it is a linear

combination of the 2 vectors and .→B

→C

××→→→CBA

The cross product is perpendicular to

the plane which contains and .

→→×CB

→B

→C


74

No general definition of such an operation

exists in the vector space.

Division of a vector by a vector

The only case where such a division is defined

is when the two vectors are parallel (or

antiparallel). That’s why we do not have a

general definition of this operation.

phys201 11314 chapter1 vectoralgebra -...

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