ENTC 3331

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ENTC 3331 RF FUNDAMENTALS
Fall 2006
Chapter 3
Vector Analysis
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Vector Analysis
1- Real number (one variable)
Scalar Quantity (mass, speed, voltage, current
and power)
2- Complex number (two variables)
Magnitude direction
Vector Algebra (velocity, electric field and
magnetic field)
Specified by one of the following coordinates
best applied to application
1- Cartesian (rectangular) 2- Cylindrical 3-
Spherical
Page 101
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Conventions

• Vector quantities denoted as or v
• We will use column format vectors
• Each vector is defined with respect to a set of
  basis vectors (which define a co-ordinate system).

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Page 101-102
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Vectors

• Vectors represent directions
• points represent positions
• Both are meaningless without reference to a
  coordinate system
• vectors require a set of basis vectors
• points require an origin and a vector space

both vectors equal
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Co-ordinate Systems

• Until now you have probably used a Cartesian
  basis
• basis vectors are mutually orthogonal and unit
  length
• basis vectors named x, y and z
• We need to define the relationship between the 3
  vectors.

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Cartesian Coordinate System
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Vector Magnitude

• The magnitude or norm of a vector of dimension n
  is given by the standard Euclidean distance
  metric
• For example
• Vectors of length 1(unit) are normal vectors.

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Normal Vectors

• When we wish to describe direction we use
  normalized vectors.
• We often need to normalize a vector

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Vector Addition and Subtraction
Subtraction
Addition
R12
Page 103
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Vector Addition
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Vector Addition

• Addition of vectors follows the parallelogram law
  in 2D and the parallelepiped law in higher
  dimensions

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Vector Subtraction
v-u
u
v
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Problem 3-1

• Vector starts at point (1,-1,-2) and ends at
  point (2,-1,0). Find a unit vector in the
  direction of .

z
y
x
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Problem 3-1
z
y
x
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Vector Multiplication
1- Simple Product 2- Scalar or Dot Product 3-
Vector or Cross Product
1- Simple Product
Scalar or Dot Product
Page 104
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Vector Multiplication by a Scalar

• Each vector has an associated length
• Multiplication by a scalar scales the vectors
  length appropriately (but does not affect
  direction)

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Dot Product

• Dot product (inner product) is defined as

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Dot Product

• Note
• Therefore we can redefine magnitude in terms of
  the dot-product operator
• Dot product operator is commutative
  distributive.

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Problem 3.2

• Given vectors
• show that is perpendicular to both and
  .

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Recall
Also, recall
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Similarly
If the angle between the two vectors is 90?.
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Problem 3.2
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Page 105
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Vector or Cross Product
Page 105-106
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Show
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Recall
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Problem 3.3

• In the Cartesian coordinate system, the three
  corners of a triangle are P1(0,2,2), P2(2,-2,2),
  and P3(1,1,-2). Find the area of the triangle.

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z
y
x
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z
y
x
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z
y
x
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z
y
x
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z
y
x
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z
y
x
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z
y
x
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z
y
x
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z
y
x
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z
y
x
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z
y
x
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z
y
x
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z
y
x
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z
y
x
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Let
?
and
?
represent two sides of the triangle. Since the
magnitude of the cross product is the area of the
parallelogram, half of this magnitude is the area
of the triangle.
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• The area of the triangle is 9 sq. units.

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Scalar and Vector Triple Products
Scalar Triple Product
Vector Triple Product
Page 107-108
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Summary
Addition
Subtraction
Vector Multiplication
1- Simple Product 2- Scalar or Dot Product 3-
Vector or Cross Product
Scalar Triple Product
Vector Triple Product
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Orthogonal Coordinate Systems (coordinates
mutually perpendicular)
z
P(x,y,z)
Cartesian Coordinates
y
P (x,y,z)
x
z
z
P(r, ?, z)
Cylindrical Coordinates
P (r, T, z)
y
r
x
?
z
P(r, ?, F)
Spherical Coordinates
?
r
P (r, T, F)
y
F
x
Page 108
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• Parabolic Cylindrical Coordinates (u,v,z)
• Paraboloidal Coordinates (u, v, F)
• Elliptic Cylindrical Coordinates (u, v, z)
• Prolate Spheroidal Coordinates (?, ?, f)
• Oblate Spheroidal Coordinates (?, ?, f)
• Bipolar Coordinates (u,v,z)
• Toroidal Coordinates (u, v, F)
• Conical Coordinates (?, µ, ?)
• Confocal Ellipsoidal Coordinate (?, µ, ?)
• Confocal Paraboloidal Coordinate (?, µ, ?)

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z
P(x,y,z)
y
x
Cartesian Coordinates P(x,y,z)
Cylindrical Coordinates P(r, ?, z)
Spherical Coordinates P(r, ?, F)
z
z
z
P(r, ?, z)
P(r, ?, F)
?
r
y
r
x
y
?
F
x
forward
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Cartesian Coordinates
( x, y, z)
z
Vector representation
z1
Z plane
x plane
Magnitude of A
y plane
Az
y1
y
Ay
Ax
x1
Position vector A
x
Base vector properties
Page 109
Back
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Cartesian Coordinates
z
Dot product
Az
y
Ay
Cross product
Ax
x
Back
Page 108
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Cartesian Coordinates
Page 109
Back
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Cylindrical Coordinates
( r, ?, z)
A1
Vector representation
Base Vectors
Magnitude of A
Base vector properties
Position vector A
Pages 109-112
Back
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Cylindrical Coordinates
Dot product
A
B
Cross product
Back
Pages 109-111
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Cylindrical Coordinates
Differential quantities Length Area Vo
lume
Pages 109-112
Back
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Spherical Coordinates
(R, ?, F)
Vector representation
Magnitude of A
Position vector A
Base vector properties
Pages 113-115
Back
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Spherical Coordinates
Dot product
A
B
Cross product
Back
Pages 113-114
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Spherical Coordinates
Differential quantities Length Area
Volume
Pages 113-115
Back
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Cartesian to Cylindrical Transformation
Back
Page 115
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Convert the coordinates of P1(1,2,0) from the
Cartesian to the Cylindrical and Spherical
coordinates.
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Convert the coordinates of P1(1,2,0) from the
Cartesian to the Cylindrical coordinates.
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Convert the coordinates of P1(1,2,0) from the
Cartesian to the Spherical coordinates.
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Convert the coordinates of P3(1,1,2) from the
Cartesian to the Cylindrical and Spherical
coordinates.
z
y
x
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Convert the coordinates of P3(1,1,2) from the
Cartesian to the Cylindrical coordinates.
z
y
x
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Convert the coordinates of P3(1,1,2) from the
Cartesian to the Spherical coordinates.
z
y
x
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