3. Coordinate Systems
Used to describe the position of a point in space
Coordinate system consists of
A fixed reference point called the origin
Specific axes with scales and labels
Instructions on how to label a point relative to the origin and the axes
4. Cartesian Coordinate System
Also called rectangular
coordinate system
x- and y- axes intersect
at the origin
Points are labeled (x,y)
5. Polar Coordinate System
Origin and reference line
are noted
Point is distance r from
the origin in the direction
of angle , ccw from
reference line
Points are labeled (r,)
6. Polar to Cartesian Coordinates
Based on forming
a right triangle
from r and
x = r cos
y = r sin
7. Trigonometry Review
Given various radius vectors,
find
Length and angle
x- and y-components
Trigonometric functions: sin, cos,
tan
8. Cartesian to Polar Coordinates
r is the hypotenuse and
an angle
must be ccw from positive
x axis for these equations to
be valid
2 2
tan
y
x
r x y
9. Example 1.1
The Cartesian coordinates of a
point in the xy plane are (x,y) =
(-3.50, -2.50) m, as shown in
the figure. Find the polar
coordinates of this point.
Solution:
2 2 2 2
( 3.50 m) ( 2.50 m) 4.30 m
r x y
2.50 m
tan 0.714
3.50 m
216 (signs give quadrant)
y
x
10. Example 1.1, cont.
Change the point in the x-y plane
Note its Cartesian coordinates
Note its polar coordinates
Please insert
active fig. 3.3
here
11. Vectors and Scalars
Vector analysis is a mathematical tool with which electromagnetic
(EM) concepts are most conveniently expressed and best
comprehended.
A scalar quantity is completely specified by a single value with an
appropriate unit and has no direction. (e.g.,mass, temperature, electric
potential, population). it has only magnitude
A vector quantity is completely described by a number and appropriate units
plus a direction. It has both magnitude and direction (e.g., Velocity, force,
electric field intensity).
Vectors are typically illustrated by drawing an ARROW above the symbol.
The arrow is used to convey direction and magnitude.
12. Vector Example
A particle travels from A to B
along the path shown by the
dotted red line
This is the distance
traveled and is a scalar
The displacement is the
solid line from A to B
The displacement is
independent of the path
taken between the two
points
Displacement is a vector
13. Vector Notation
Text uses bold with arrow to denote a vector:
Also used for printing is simple bold print: A
When dealing with just the magnitude of a
vector in print, an italic letter will be used: A or
| |
The magnitude of the vector has physical units
The magnitude of a vector is always a positive
number
When handwritten, use an arrow:
A
A
A
14. Equality of Two Vectors
Two vectors are equal if they
have the same magnitude and
the same direction
if A = B and they point
along parallel lines
All of the vectors shown are
equal
A B
15. Adding Vectors
When adding vectors, their directions must be taken into
account
Units must be the same
Graphical Methods
Use scale drawings
Algebraic Methods
More convenient
16. Adding Vectors Graphically
Choose a scale
Draw the first vector, , with the appropriate length and in the
direction specified, with respect to a coordinate system
Draw the next vector with the appropriate length and in the direction
specified, with respect to a coordinate system whose origin is the
end of vector and parallel to the coordinate system used for
A
A
A
17. Adding Vectors Graphically, cont.
Continue drawing the
vectors “tip-to-tail”
The resultant is drawn
from the origin of to
the end of the last
vector
Measure the length of
and its angle
Use the scale factor to
convert length to actual
magnitude
A
R
18. Adding Vectors Graphically, final
When you have many
vectors, just keep
repeating the process
until all are included
The resultant is still
drawn from the tail of
the first vector to the tip
of the last vector
19. Adding Vectors, Rules
When two vectors are added, the
sum is independent of the order
of the addition.
This is the Commutative Law of
Addition
A B B A
20. Adding Vectors, Rules cont.
When adding three or more vectors, their sum is independent of
the way in which the individual vectors are grouped
This is called the Associative Property of Addition
A B C A B C
21. Adding Vectors, Rules final
When adding vectors, all of the vectors must have the same
units
All of the vectors must be of the same type of quantity
For example, you cannot add a displacement to a velocity
22. Negative of a Vector
The negative of a vector is defined as the vector that, when
added to the original vector, gives a resultant of zero
Represented as
The negative of the vector will have the same magnitude, but
point in the opposite direction
A
0
A A
23. Subtracting Vectors
Special case of vector addition
If , then use
Continue with standard vector
addition procedure
A B
A B
24. Subtracting Vectors, Method 2
Another way to look at
subtraction is to find the vector
that, added to the second vector
gives you the first vector
As shown, the resultant vector
points from the tip of the second to
the tip of the first
A B C
25. Multiplying or Dividing a Vector by a
Scalar
The result of the multiplication or division of a vector by a scalar is a
vector
The magnitude of the vector is multiplied or divided by the scalar
If the scalar is positive, the direction of the result is the same as of the
original vector
If the scalar is negative, the direction of the result is opposite that of the
original vector
26. Component Method of Adding Vectors
Graphical addition is not recommended when
High accuracy is required
If you have a three-dimensional problem
Component method is an alternative method
It uses projections of vectors along coordinate axes
27. Components of a Vector, Introduction
A component is a
projection of a vector along
an axis
Any vector can be completely
described by its components
It is useful to use
rectangular components
These are the projections of
the vector along the x- and y-
axes
28. Vector Component Terminology
are the component vectors of
They are vectors and follow all the rules for vectors
Ax and Ay are scalars, and will be referred to as the
components of
x y
and
A A
A
A
29. Components of a Vector
Assume you are given a vector
It can be expressed in terms of
two other vectors, and
These three vectors form a right
triangle
A
x
A y
A
x y
A A A
30. Components of a Vector, 2
The y-component is
moved to the end of
the x-component
This is due to the fact
that any vector can be
moved parallel to
itself without being
affected
This completes the
triangle
31. Components of a Vector, 3
The x-component of a vector is the projection along the x-axis
The y-component of a vector is the projection along the y-axis
This assumes the angle θ is measured with respect to the x-axis
If not, do not use these equations, use the sides of the triangle directly
cos
x
A A
sin
y
A A
32. Components of a Vector, 4
The components are the legs of the right triangle whose hypotenuse is
the length of A
May still have to find θ with respect to the positive x-axis
2 2 1
and tan y
x y
x
A
A A A
A
33. Components of a Vector, final
The components can be
positive or negative and
will have the same units
as the original vector
The signs of the
components will depend
on the angle
34. Unit Vectors
A unit vector is a dimensionless vector with a magnitude of
exactly 1.
Unit vectors are used to specify a direction and have no other
physical significance
35. Unit Vectors, cont.
The symbols
represent unit vectors
They form a set of mutually
perpendicular vectors in a right-
handed coordinate system
Remember,
k̂
and
,
ĵ
,
î
ˆ ˆ ˆ 1
i j k
36. Viewing a Vector and Its Projections
Rotate the axes for various
views
Study the projection of a vector
on various planes
x, y
x, z
y, z
37. Unit Vectors in Vector Notation
Ax is the same as Ax
and Ay is the same as
Ay etc.
The complete vector
can be expressed as
î
ĵ
ˆ ˆ
x y
A A
A i j
38. Adding Vectors Using Unit Vectors
Using
Then
and so Rx = Ax + Bx and Ry = Ay + By
ˆ ˆ ˆ ˆ
ˆ ˆ
ˆ ˆ
x y x y
x x y y
x y
A A B B
A B A B
R R
R i j i j
R i j
R i j
2 2 1
tan y
x y
x
R
R R R
R
R A B
39. Adding Vectors with Unit Vectors
Note the relationships among the
components of the resultant and
the components of the original
vectors
Rx = Ax + Bx
Ry = Ay + By
40. Three-Dimensional Extension
Using
Then
and so Rx= Ax+Bx, Ry= Ay+By, and Rz =Az+Bz
ˆ ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ
ˆ ˆ ˆ
x y z x y z
x x y y z z
x y z
A A A B B B
A B A B A B
R R R
R i j k i j k
R i j k
R i j k
2 2 2 1
cos , .
x
x y z
R
R R R R etc
R
R A B
41.
r x y z
x y z
x y z
r a a a
, , :
x y z
a a a unit vectors
?
PQ
R
PQ Q P
R r r
(2 2 ) (1 2 3 )
x y z x y z
a a a a a a
4 2
x y z
a a a
Vector Components and Unit Vectors
42. For any vector B, :
x x y y z z
B B B
B a a + a
2 2 2
x y z
B B B
B Magnitude of B
B
2 2 2
B
x y z
B B B
B
a
B
B Unit vector in the direction of B
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
Vector Components and Unit Vectors
43. • Differential surface units:
dx dy
dy dz
dx dz
• Differential volume unit :
dx dy dz
Rectangular Coordinate System
44. 44
The three basic laws of algebra obeyed by any given vector
A, B, and C, are summarized as follows:
Law Addition Multiplication
Commutative
Associative
Distributive
where k and l are scalars
A
B
B
A
C
)
B
A
(
)
C
B
(
A
k
A
A
k
A
)
kl
(
)
A
l
(
k
B
k
A
k
)
B
A
(
k
45. 45
When two vectors and are multiplied, the result is
either a scalar or a vector depending on how they are
multiplied. There are two types of vector multiplication:
1. Scalar (or dot) product:
2.Vector (or cross) product:
The dot product of the two vectors and is defined
geometrically as the product of the magnitude of and the
projection of onto (or vice versa):
where is the smaller angle between and
B
A B
A B
B
A
B
A
A B
cos
AB
B
A
A B
A B
A B
Multiplication of Vectors:
46. Given two vectors A and B, the dot product, or scalar product, is defines as the
product of the magnitude of A, the magnitude of B, and the cosine of the smaller
angle between them:
cos AB
A B A B
The dot product is a scalar, and it obeys the commutative law:
A B B A
For any vector and ,
x x y y z z
A A A
A a a + a x x y y z z
B B B
B a a + a
x x y y z z
A B A B A B
A B +
The Dot Product
47. 47
If and then
which is obtained by multiplying and component by
component
)
,
A
,
A
,
A
(
A Z
Y
X
)
B
,
B
,
B
(
B Z
Y
X
Z
Z
Y
Y
X
X B
A
B
A
B
A
B
A
A B
A
B
B
A
C
A
B
A
C
B
A
)
(
A A A
2
A2
eX ex ey ey eZ ez 1
eX ey ey ez eZ ex 0
48.
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
Example
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) the angle
θBAC at vertex A; (d) the vector projection of RAB on RAC.
The Dot Product
49. One of the most important applications of the dot product is that of
finding the component of a vector in a given direction.
Chapter 1
cos Ba
B a B a cos Ba
B
• The scalar component of B in the direction
of the unit vector a is Ba
• The vector component of B in the direction
of the unit vector a is (Ba)a
The Dot Product
50. Chapter 1 Vector Analysis
Example
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) the angle
θBAC at vertex A; (d) the vector projection of RAB on RAC.
( 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
The Dot Product
51. sin
N AB
A B a A B
Chapter 1 Vector Analysis
Given two vectors A and B, the magnitude of the cross product, or vector product,
written as AB, is defines as the product of the magnitude of A, the magnitude of B,
and the sine of the smaller angle between them.
The direction of AB is perpendicular to the plane containing A and B and is in the
direction of advance of a right-handed screw as A is turned into B.
The cross product is a vector, and it is not
commutative:
( ) ( )
B A A B
x y z
y z x
z x y
a a a
a a a
a a a
The Cross Product
52. 52
The cross product of two vectors and is defined as
where is a unit vector normal to the plane containing
and . The direction of is determined using the right-
hand rule or the right-handed screw rule.
A
A
n
A Be
sin
AB
B
A
B
B
n
e
n
e
B
A
Direction of
and using
(a) right-hand rule,
(b) right-handed
screw rule
n
e
54. Chapter 1
Example
Given A = 2ax–3ay+az and B = –4ax–2ay+5az, find AB.
( ) ( ) ( )
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
𝑨 × 𝑩 =
𝐚𝑥 𝐚𝑦 𝐚𝑧
𝐴𝑥 𝐴𝑦 𝐴𝑧
𝐵𝑥 𝐵𝑦 𝐵𝑧
The Cross Product
55. 55
Note that the cross product has the following basic
properties:
(i) It is not commutative:
It is anticommutative:
(ii) It is not associative:
(iii) It is distributive:
(iv)
A
B
B
A
A
B
B
A
C
)
B
A
(
)
C
B
(
A
C
A
B
A
C
B
A
)
(
0
A
A
)
0
(sin
56. 56
Also note that
which are obtained in cyclic permutation and illustrated
below.
y
x
z
x
z
y
z
y
x
e
e
e
e
e
e
e
e
e
Cross product using cyclic permutation: (a) moving clockwise leads to positive results;
(b) moving counterclockwise leads to negative results
57. 57
Scalar and Vector Fields
• A field can be defined as a function that specifies a particular quantity
everywhere in a region (e.g., temperature, distribution in a building), or as a
spatial distribution of a quantity, which may or may not be a function of time.
Scalar quantity scalar function of position scalar field
Vector quantity vector function of position vector field
• A field (scalar or vector) may be defined mathematically as some function of
that vector which connects an arbitrary origin to a general point in space.
• The temperature throughout the bowl of soup and the density at any point in
the earth are examples of scalar fields .
• The gravitational and magnetic fields of the earth, the voltage gradient in a
cable are examples of vector fields .
59. 59
• In order to define the position of a point in space, an appropriate
coordinate system is needed.
• A considerable amount of work and time may be saved by choosing a
coordinate system that best fits a given problem.
• A hard problem in one coordinate system may turn out to be easy in
another system.
• We will consider the Cartesian, the circular cylindrical, and the
spherical coordinate systems.
• All three are orthogonal(the coordinates are mutually perpendicular).
The Coordinate Systems
60. 60
Cartesian coordinates (x,y,z)
The ranges of the coordinate variables are
A vector in Cartesian coordinates can be written as
The intersection of three
orthogonal infinite places
(x=const, y= const, and z =
const)
defines point P.
z
y
x
z
z
y
y
x
x
z
y
x e
A
e
A
e
A
or
)
A
,
A
,
A
(
A
Constant x, y and z surfaces
The Cartesian Coordinate System
63. Cylindrical Coordinates .
- the radial distance from the z – axis
- the azimuthal angle, measured from the x-
axis in the xy – plane
- the same as in the Cartesian system.
A vector A in cylindrical coordinates can be written as
Cylindrical coordinates amount to a combination of
rectangular coordinates and polar coordinates.
)
z
,
,
(
z
2
0
0
2
/
1
2
2
2
)
A
(
A
A
or
)
,
(
z
z
z
z
A
A
e
A
e
A
e
A
A
A
The Cylindrical Coordinate System
65. Chapter 1
• Differential surface units:
d dz
d dz
d d
• Differential volume unit :
d d dz
cos
x
sin
y
z z
2 2
x y
1
tan
y
x
z z
• Relation between the rectangular
and the cylindrical coordinate
systems
The Cylindrical Coordinate System
66. 66
Positions in the x-y plane are determined by the values of
Relationship between (x,y,z) and )
z
,
,
(
and
z
z
x
y
tan
y
x 1
2
2
Review of Vector Analysis
69. 69
Differential elements in cylindrical coordinates
Metric coefficient
z
p a
dz
a
d
a
d
dl
dz
d
d
dv
Review of Vector Analysis
70.
a
z
a
a
Chapter 1
• 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
The Cylindrical Coordinate System
72. Spherical coordinates .
- the distance from the origin to the point P
- the angle between the z-axis and the radius
vector of P
- the same as the azimuthal angle in
cylindrical coordinates
)
,
,
r
(
0 r
0
Colatitude
(polarangle)
0 2
The Spherical Coordinate System
74. Chapter 1 Vector Analysis
• Differential surface units:
dr rd
sin
dr r d
sin
rd r d
• Differential volume unit :
sin
dr rd r d
The Spherical Coordinate System
75. sin cos
x r
sin sin
y r
cos
z r
2 2 2
, 0
r x y z r
1
2 2 2
cos , 0 180
z
x y z
1
tan
y
x
• Relation between the rectangular and the
spherical coordinate systems
Chapter 1
• Dot products of unit vectors in spherical and
rectangular coordinate systems
The Spherical Coordinate System
77. Example
Given the two points, C (–3,2,1) and D (r = 5, θ = 20°, Φ = –70°), find: (a) the
spherical coordinates of C; (b) the rectangular coordinates of D.
2 2 2
r x y z
1
2 2 2
cos
z
x y z
1
tan
y
x
2 2 2
( 3) (2) (1)
3.742
1 1
cos
3.742
74.50
1 2
tan
3
33.69 180
146.31
( 3.742, 74.50 , 146.31 )
C r
( 0.585, 1.607, 4.698)
D x y z
The Spherical Coordinate System
79. 79
Differential elements in the spherical coordinate system
a
d
sin
r
a
rd
a
dr
dl r
d
drd
sin
r
dv 2
Review of Vector Analysis
