The document summarizes key concepts from Chapter 1 of the textbook "Engineering Electromagnetics - 8th Edition" by William H. Hayt, Jr. & John A. Buck. It introduces scalar and vector quantities, describes vector algebra including addition, subtraction and multiplication. It also discusses various coordinate systems used to describe the location and direction of vectors including rectangular, cylindrical and spherical coordinate systems. Transformations between Cartesian and other coordinate systems are shown.

1. Text Book
Engineering Electromagnetics – 8th Edition
 William H. Hayt, Jr. & John A. Buck

14. Chapter 1
Vector Analysis

15. Electric field
Magnetic field
Produced by the motion of
electric charges, or electric
current, and gives rise to the
magnetic force associated
with magnets.
Electromagnetic is the study of the effects of charges at rest
and charges in motion
Produced by the presence of
electrically charged particles,
and gives rise to the electric
force.

16. Scalars &Vectors
Scalar
 Refers to a Quantity whose value may be
represented by a Single (Positive/Negative)
real number.
 Body falling a distance L in Time t
 TemperatureT at any point in a bowl of soup
whose co-ordinates are x, y, z
 L, t,T, z, y & z are all scalars
 Mass, Density, Pressure,Volume,Volume
resistivity,Voltage

17. Scalars &Vectors
Vector
 A quantity who has both a magnitude and
direction in space.
 Force,Velocity,Acceleration & a straight line
from positive to negative terminal of a storage
battery

18. Vector Algebra
Addition
Associative Law:
Distributive Law:

19. Vector Algebra
Coplanar vectors
 Lying in a common plane

20. Vector Algebra
Subtraction
 A – B = A + (-B)
Multiplication
 Obeys Associative & Distributive laws
 (r + s) (A + B) = r(A + B) + s (A + B)
= rA + rB + sA + sB

21. Orthogonal Coordinate
Systems
A coordinate system defines a set of reference
directions. In a 3D space, a coordinate system can
be specified by the intersection of 3 surfaces at
each and every point in space.
The origin of the coordinate system is the
reference point relative to which we locate every
other point in space.

22. Orthogonal Coordinate
Systems
A position vector defines the position of a point
in space relative to the origin.
These three reference directions are referred to
as coordinate directions or base vectors, and
are usually taken to be mutually perpendicular
(orthogonal) . In this class, we use three coordinate
systems:
 Cartesian
 Cylindrical
 Spherical

23. Rectangular Coordinate System
In Cartesian or rectangular coordinate system a
point P is represented by coordinates (x,y,z) All the
three coordinates represent the mutually
perpendicular plane surfaces
The range of coordinates are
-∞< x< ∞
-∞< y< ∞
-∞< z< ∞

25. Point Locations in Rectangular
Coordinates

26. Differential Volume Element

27. Orthogonal Vector Components

28. Orthogonal Unit Vectors

29. Vector Representation in Terms of
Orthogonal Rectangular Components

30. Vector Expressions in Rectangular
Coordinates
General Vector, B:
Magnitude of B:
Unit Vector in the
Direction of B:

31. Example

32. Vector Components and Unit
Vectors
Example
• Given points M(–1,2,1) and N(3,–3,0), find RMN and aMN.
(3 3 0 ) ( 1 2 1 )
MN x y z x y z
      
R a a a a a a 4 5
x y z
  
a a a
MN
MN
MN

R
a
R 2 2 2
4 5 1
4 ( 5) ( 1)
x y z
 

   
a a a
0.617 0.772 0.154
x y z
  
a a a

33. Field
Function, which specifies a particular
quantity everywhere in the region
Two types:
 Vector Field has a direction feature
pertaining to it e.g. Gravitational field in
space and
 Scalar Field has only magnitude e.g.
Temperature

34. The Dot Product
Commutative Law:

35. Vector Projections Using the Dot Product
B • a gives the component of B
in the horizontal direction
(B • a) a gives the vector component
of B in the horizontal direction

36. Operational Use of the Dot Product
Given
Find
where we have used:
Note also:

37. The three vertices of a triangle are located at A(6,–1,2),
B(–2,3,–4), and C(–3,1,5). Find: (a) RAB; (b) RAC; (c) angle
θBAC at vertex A; (d) the vector projection of RAB on RAC.
Drill Problem

38. Solution
( 2 3 4 ) (6 2 )
AB x y z x y z
      
R a a a a a a 8 4 6
x y z
   
a a a
( 3 1 5 ) (6 2 )
AC x y z x y z
      
R a a a a a a 9 2 3
x y z
   
a a a
A
B
C
BAC

cos
AB AC AB AC BAC

 
R R R R
cos AB AC
BAC
AB AC


 
R R
R R 2 2 2 2 2 2
( 8 4 6 ) ( 9 2 3 )
( 8) (4) ( 6) ( 9) (2) (3)
x y z x y z
      

      
a a a a a a
62
116 94

1
cos (0.594)
BAC
 
  53.56
 
0.594


39. Continued …
 
on
AB AC AB AC AC
 
R R R a a
2 2 2 2 2 2
( 9 2 3 ) ( 9 2 3 )
( 8 4 6 )
( 9) (2) (3) ( 9) (2) (3)
x y z x y z
x y z
 
     
 
   
 
     
 
 
a a a a a a
a a a
( 9 2 3 )
62
94 94
x y z
  

a a a
5.963 1.319 1.979
x y z
   
a a a

40. Cross Product

41. Operational Definition of the Cross Product in
Rectangular Coordinates
Therefore:
Or…
Begin with:
where

42. Operational Definition of the Cross
Product in Rectangular Coordinates

43. The Cross Product
Example
Given A = 2ax–3ay+az and B = –4ax–2ay+5az, find AB.
( ) ( ) ( )
y z z y x z x x z y x y y x z
A B A B A B A B A B A B
      
A B a a a
     
( 3)(5) (1)( 2) (1)( 4) (2)(5) (2)( 2) ( 3)( 4)
x y z
           
a a a
13 14 16
x y z
   
a a a

44. Coordinate Systems
Purpose
 To uniquely determine the position of an object
or data point in space.
 By ‘space’ we may literally mean in physical
space, but in general it simply refers to what we
might call ‘variable-space’, where each dimension
corresponds to one variable.
 A graph of stock prices probably has variables
of ‘time’ and ‘value’, so we are in time-value
space, and our coordinate system and any
equations we might write need to specify the
time and value of each data point.

45. Common Coordinate Systems
Number line
 In this system, an arbitrary point O (the
origin) is chosen on a given line.
 The coordinate of a point P is defined as the
signed distance from O to P, where the signed
distance is the distance taken as positive or
negative depending on which side of the line P
lies.
 Each point is given a unique coordinate and
each real number is the coordinate of a
unique point.

46. Common Coordinate Systems
Cartesian Coordinate System
 In the plane, two perpendicular lines are
chosen and the coordinates of a point are
taken to be the signed distances to the lines.

47. Common Coordinate Systems
Polar Coordinate System
 For some applications including many curves,
rotations, and complex numbers, it is simpler
to use a coordinate system based on the
circle.
 These are polar coordinates, and our two
parameters are r, the radial distance between
the point and the origin, and θ, the angle
between the point and the positive x-axis

48. Common Coordinate Systems
Cylindrical Coordinates
 If you are familiar with 3D Cartesian coordinates
and 2D polar coordinates, cylindrical coordinates
are a very easy extension.
 They specify a radial distance r and an angle  in
one plane, as for polar coordinates, and a
distance z perpendicular to that plane.
 The relationship between cylindrical coordinates
and Cartesian ones is identical to that between
polar and Cartesian, with the addition of z = z.

49. Common Coordinate Systems
Relationship between Cylindrical and
Cartesian Coordinate System

50. Circular Cylindrical Coordinates
Point P has coordinates
Specified by P(z)
The ρ coordinate represents a
cylinder of radius ρ with z axis as
its axis. The ø coordinate (the
azimuthal angle) is measured from
x axis in xy plane. Z is same as in
Cartesian coordinates
The range of coordinates are
0≤ ρ < ∞
0≤ ø < 2π
-∞ < z < ∞

51. Differential Volume in Cylindrical
Coordinates
dv = dddz

52. Point Transformations in Cylindrical
Coordinates

53. 
a
z
a

a
The Cylindrical Coordinate System
Dot products of unit vectors in
cylindrical and rectangular coordinate
systems
y
a
z
a
x
a
A 
 
A a
( )
x x y y z z
A A A 
  
a a + a a
x x y y z z
A A A
  
    
a a a a + a a
cos sin
x y
A A
 
 
A 
 
A a
( )
x x y y z z
A A A 
  
a a + a a
x x y y z z
A A A
  
    
a a a a + a a
sin cos
x y
A A
 
  
z z
A  
A a
( )
x x y y z z z
A A A
  
a a + a a
x x z y y z z z z
A A A
    
a a a a + a a
z
A

?
x x y y z z z z
A A A A A A
   
    
A a a + a A a a + a

54. Transform the vector B into cylindrical coordinates
Start with:
Problem

55. Then:

56. Finally:

57. Common Coordinate Systems
Spherical Coordinates
 Can be viewed as a 3D extension of polar
Coordinates
 In this case, the third parameter is another
angle, , measured from the `north pole', and
r refers to the total distance of the point
from the origin, not the distance in one plane.
 The earth's lines of latitude and longitude are
a familiar system of spherical coordinates.
 Longitude is the , spanning 360 degrees or
2 radians, latitude is the  spanning 180
degrees or  radians, and r is assumed to be
radius of the earth.

58. Common Coordinate Systems
Point Transformation

59. The r coordinate represents a sphere
of radius r centered at origin . The θ
coordinate represents the angle made
by the cone with z-axis.
The ø coordinate is the same as
cylindrical coordinate.
The range of coordinates are
0≤ r < ∞
0≤ θ ≤ π
0≤ ø < 2π
Point P has coordinates
Specified by P(r)
Spherical Coordinates