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ELECTROMAGNETIC FIELD WAVE
LECTURE-2
Vector Analysis
The quantities that we deal in electromagnetic theory may be either scalar or vectors [There are
other classes of physical quantities called Tensors: scalars and vectors are special cases]. Scalars
are quantities characterized by magnitude only and algebraic sign. A quantity that has direction
as well as magnitude is called a vector. Both scalar and vector quantities are function of time and
position. A field is a function that specifies a particular quantity everywhere in a region.
Depending upon the nature of the quantity under consideration, the field may be a vector or a
scalar field. Example of scalar field is the electric potential in a region while electric or magnetic
fields at any point is the example of vector field.
Fundamentals Vector Algebra:
A vector is represented by a directed line segment: length of the line is proportional to
magnitude and the orientation of the directed line segment with respect to some reference gives
the vector.
ˆ , where A  A is the magnitude and aˆ 
A vector A can be written as A  aA
vector which has unit magnitude and direction same as that of A .
Two vectors A and B are added together to give another vector C . We have:
C  A B
Figure 1: Vector addition
Vector subtraction is similarly carried out as:
D  A  B  A  ( B )
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A
is the unit
A
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Figure 2: Vector subtraction
Scaling of a vector is defined as C   B , where C is a scaled version of vector B and  is a
scalar.
Some important laws of vector algebra are:
A B  B  A
A  ( B  C )  ( A  B)  C
Commutative law
Associative law
 ( A  B)   A   B
Distributive law
The position vector rP of a point P is the directed distance from the origin ( O ) to P i.e.
rP  OP .
Figure 3: Distant vector
If rP  OP and rQ  OQ are the position vectors of the points P and Q then the distance vector
PQ  OQ  OP  rQ  rP
Product of Vectors
When two vectors A and B are multiplied, the result is either a scalar or a vector depending
how the two vectors were multiplied. The two types of vector multiplication are:
Scalar product (or dot product) A B gives a scalar
Vector product (or cross product) A  B gives a vector
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The dot product between two vectors is defined as A B  AB cos AB .
Figure: Dot product
The dot product is commutative i.e. A B  B A and distributive i.e. A ( B  C )  A B  A C .
Associative law does not apply to scalar product.
The vector or cross product of two vectors A and B is denoted by A  B . A  B is a vector
perpendicular to the plane containing A and B , the magnitude is given by ABSin AB and
direction is given by right hand rule as explained in Figure.
Figure: Vector cross product
A  B  aˆn ABSin AB , where
aˆn 
aˆ n
is the unit vector given by
A B
A B
The following relations hold for vector product.
A  B  B  A
i.e. cross product is non commutative
A (B  C)  A B  A C
i.e. cross product is distributive
A  ( B  C )  ( A  B)  C
i.e. cross product is non associative
Scalar and vector triple product
Scalar triple product
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A ( B  C )  B (C  A)  C ( A  B)
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Vector triple product A  ( B  C )  B( A C )  C ( A B)
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