Vector

Electromagnetic Field Theory
Unit-1
Vectors
1
Mohammad Akram,Assistant Professor,ECE
Department

Introduction
Electromagnetics (EM)is a branch of physics or electrical engineering
in which electric and magnetic phenomena are studied.
• EM principles find applications in various allied disciplines
such as
1. Microwaves
2. Antennas
3. Electric machines
4. Satellite communications
5. Nuclear research
6. Fiber optics
7. Electromagnetic interference and compatibility
8. Electromechanical energy conversion
9. Radar meteorology
10. Remote sensing etc
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•In Physical Medicine: EM power, either in the form of
shortwaves or microwaves, is used to heat deep tissues and to
stimulate certain physiological responses in order to relieve
certain pathological conditions
•In Induction Heaters: EM fields are used in induction heaters
for melting, forging, annealing, surface hardening, and soldering
operations. Dielectric heating equipment uses shortwaves to join
or seal thin sheets of plastic materials.
•In Agriculture: It is used, for example, to change vegetable taste
by reducing acidity.
•EM devices include transformers, electric relays, radio/TV,
telephone, electric motors, transmission lines, waveguides,
antennas, optical fibers, radars, and lasers. The design of
these devices requires thorough knowledge of the laws and
principles of EM. 3
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Scalar and Vector
Scalar: A scalar is a quantity that has only magnitude.
• Quantities such as time, mass, distance, temperature, entropy,
electric potential, and population are scalars
Vector: The quantities which have magnitude as well as direction are
termed as vector quantities.
• Examples are velocity, acceleration, force, momentum etc.
• In fig.(1),OA is the vector. Point O is known as origin or tail and
point A is known as terminus or head.
• Vectors are denoted by letters in bold faces (OA) or by putting
arrow on the letters(OA)
AO
Fig.(1)
OA
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Addition of Vectors
• Now we will learn the addition method of two vectors A and B of
magnitudes A and B.
• First of all we draw a line OP of magnitude A and put an arrow
at P. Now OP represents vector A. From the head of A i.e.,
from P, we draw a vector B. Now the line joining the tail of A
to the head of B is the resultant of two vectors A and B. It is
represented by vector OQ in fig.(2).
A
B
Fig.(2)
A
O P
B
Q
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•Symbolically, addition of two vectors can be represented
as R=A+B
•Let A and B be two factors acting at a point O making an angle θ
as shown in fig.(3)
•Now we set the vector B at the end of vector A and join OQ. Then
OQ (vector R) represents the sum of two vectors A and B both in
magnitude and direction.
Q
B
α
R
O P
θ
A
B
Fig.(3)
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•Magnitude of R is given by
•Let α be the angle between A and R, then
 cos2
22
ABR BA 



cos
sin
tan
BA
A


Subtraction of Vectors:
•The subtraction of the vector B from A is equivalent to the
addition of (-B) with A, i.e.,
A-B = A+(-B)
•The negative sign means a vector of equal magnitude but
opposite direction.
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The process is illustrated in the figure shown
below:
•If two vectors A and B acting at a point and making
an angle θ then magnitude of the resultant R (A-B) is
given by:
A
B
-B
A
-B
Fig.(4)
 cos2
22
ABR BA 
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UNIT VECTOR:
•A vector A has both magnitude and direction. The
magnitude of A is a scalar written as A or |A|. A unit vector
aA along A is defined as a vector whose magnitude is unity
(i.e., 1) and its direction is along A, that is,
Note that |aA| = 1. Thus we may write A as
A = A aA
which completely specifies A in terms of its magnitude A and
its direction aA.
A
A
A
A
aA

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•A vector A in Cartesian (or rectangular) coordinates
may be represented as
(Ax, Ay, Az) or Ax ax + Ay ay + Az az
where Ax, Ay and Az are called the components of A in the x,
y, and z directions respectively; ax, ay and az are unit vectors
in the x, y, and z directions, respectively.
Fig.5 (a) Unit vectors ax, ay, and az (b) components of A along ax, ay and az.
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•The magnitude of vector A is given by
and the unit vector along A is given by
POSITION VECTOR:
•A point P in Cartesian coordinates may be represented by
(x, y, z).
•The position vector rp (or radius vector) of point P is as the
directed distance from the origin O to P: i.e..
 AAA zyx
A
222

 AAA
aAaAaA
zyx
zzyyxx
222


aA
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rP=OP=xax +yay + zaz
•The position vector of point P is useful in defining its
position in space. Point (3, 4, 5), for example, and its
position vector rP=3ax +4ay + 5az are shown in Figure 6.
DISTANCE VECTOR:
•The distance vector is the displacement from one point to
another.
Fig.6 Illustration of position vector rP=3ax +4ay + 5az
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Fig.7 Position vector rPQ
•If two points P and Q are given by (xP, yP, zP) and (xQ, yQ, zQ)
the distance vector (or separation vector) is the displacement
from P to Q as shown in Figure 7; that is,
rPQ= rQ - rP
= (xQ – xP)ax + (yQ – yP)ay + (zQ – zP)az
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VECTOR MULTIPLICATION:
When two vectors A and B are multiplied, the result is
either a scalar or a vector depending on how they are
multiplied. Thus there are two types of vector
multiplication:
1. Scalar (or dot) product: A • B
2. Vector (or cross) product: A × B
Multiplication of three vectors A, B and C can result in either:
3. Scalar triple product: A • (B × C)
or
4. Vector triple product: A × (B × C)
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A. Dot Product:
•The dot product of two vectors A and B, written as A • B.
is defined geometrically as the product of the magnitudes
of A and B and the cosine of the angle between them.
Thus, A • B=ABcosθAB
where θAB is the smaller angle between A and B. The result of
A • B is called either the scalar product because it is scalar,
or the dot product due to the dot sign .
•If A= (Ax, Ay, Az) and B=(Bx, By, Bz), then
A • B= AxBx + AyBy + AzBz
which is obtained by multiplying A and B component by
component.
•Two vectors A and B are said to be orthogonal (or
perpendicular) with each other if A • B = 0. 15
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Note that dot product obeys the following:
(i) Commutative law:
A .B = B.A
(ii) Distributive law:
A.(B + C) = A.B + A.C
A.A = |A|2 = A2
(iii)Also note that
ax . ax = ay . ay = az . az = 1
ax . ay = ay . az = az . ax = 0

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CROSS PRODUCT
•The cross product of two vectors A and B, written as
A × B. is a vector quantity whose magnitude is the area
of the parallelopiped formed by A and B (see Figure 8)
and is in the direction of advance of a right-handed screw
as A is turned into B as shown in fig.(9)
Fig.8 The cross product of A and B is a vector with magnitude equal to the area of the
parallelogram and direction as indicated.

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Thus, A × B = AB SinθAB an
where an is a unit vector normal to the plane
containing A and B.
•The direction of an is taken as the direction of the right
thumb when the fingers of the right hand rotate from A to
B as shown in Figure 9.
Fig.9 Direction of A × B and an using (a) right-hand rule, (b) right-handed screw rule.

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•The vector multiplication A × B is called cross product
due to the cross sign; it is also called vector product
because the result is a vector
•If A= (Ax, Ay, Az) and B=(Bx, By, Bz), then
A × B = (AyBz - AzBy)ax + (AzBx - AxBz)ay + (AxBy - AyBx)az
•Note that the cross product has the following basic
properties:
(i) It is not commutative:
A × B ≠ B × A
It is anticommutative:
A × B = -B × A
BBB
AAA
aaa
zyx
zyx
zyx
ΒΑ

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(ii) It is not associative:
A × (B × C) ≠ (A × B) × C
(iii) It is distributive:
A × ( B + C) = A × B + A × C
(iv) A × A = 0
Also note that
ax × ay = az
ay × az = ax
az × ax = ay
Fig.10 Cross product using cyclic permutation: (a) moving
clockwise leads to positive results: (b) moving counterclockwise
leads to negative results.

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Scalar Triple Product
•Given three vectors A, B and C, we define the scalar
triple product as A • (B × C) = B • (C × A) = C • (A ×
B)
obtained in cyclic permutation.
•If A= (Ax, Ay, Az) , B= (Bx, By, Bz) and C= (Cx, Cy, Cz)
then A • (B × C) is the volume of a parallelepiped having
A, B and C as edges and is easily obtained by finding the
determinant of the 3 × 3 matrix formed by A, B and C
that is,
Since the result of this vector multiplication is scalar , it is
called the scalar triple product.
CCC
BBB
AAA
zyx
zyx
zyx
(  C)ΒΑ

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Vector Triple Product:
•For vectors A, B and C , we define the vector triple
product as A × (B × C)=B(A•C) − C(A•B)
obtained using the "bac-cab" rule
Components of a Vector:
•A direct application of vector product is its use in
determining the projection (or component) of a vector in a
given direction.
•The projection can be scalar or vector.
•Given a vector A, we define the scalar component AB of A
along vector B as (see Figure 11)

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AB=AcosθAB = |A||aB|cosθAB
or AB = A.aB
•The vector component AB of A along B is simply the
scalar component AB multiplied by a unit vector along B;
that is,
AB = ABaB = (A.aB) aB
•Both the scalar and vector components of A are illustrated in
Fig.11
Fig.11 Components of A along B: (a) scalar component AB, (b) vector
component AB.

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Reference
1. “Principles of Electromagnetics” Book by “Matthew
N.O. Sadiku”, “S.V Kulkarni”

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Vector

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