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Module 1: A Crash Course in Vectors
Lecture 1: Scalar and Vector Fields
Objectives
In this lecture you will learn the following
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•
•
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Learn about the concept of field
Know the difference between a scalar field and a vector field.
Review your knowledge of vector algebra
Learn how an area can be looked upon as a vector
Define position vector and study its transformation properties under rotation.
SCALAR AND VECTOR FIELDS
This introductory chapter is a review of mathematical concepts required for the course. It is assumed
that the reader is already familiar with elementary vector analysis. Physical quantities that we deal with
in electromagnetism can be scalars or vectors.
A scalar is an entity which only has a magnitude. Examples of scalars are mass, time, distance, electric
charge, electric potential, energy, temperature etc.
A vector is characterized by both magnitude and direction. Examples of vectors in physics are
displacement, velocity, acceleration, force, electric field, magnetic field etc. A field is a quantity which
can be specified everywhere in space as a function of position. The quantity that is specified may be a
scalar or a vector. For instance, we can specify the temperature at every point in a room. The room may,
therefore, be said to be a region of ``temperature field" which is a scalar field because the temperature
is a scalar function of the position. An example of a scalar field in electromagnetism is the
electric potential. In a similar manner, a vector quantity which can be specified at every point in a
region of space is a vector field. For instance, every point on the earth may be considered to be in the
gravitational force field of the earth. we may specify the field by the magnitude and the direction of
acceleration due to gravity (i.e. force per unit mass )
at every point in space. As another
example consider flow of water in a pipe. At each point in the pipe, the water molecule has a velocity
. The water in the pipe may be said to be in a velocity field. There are several examples of
vector field in electromagnetism, e.g., the electric field
, the magnetic flux density
etc.
1

Elementary Vector Algebra: Geometrically a vector is represented by a directed line
segment. Since a vector remains unchanged if it is shifted parallel to itself, it does not have any
position information. A three dimensional vector can be specified by an ordered set of three
numbers, called its components. The magnitude of the components depends on the coordinate
system used. In electromagnetism we usually use Cartesian, spherical or cylindrical coordinate
systems. (Specifying a vector by its components has the advantage that one can extend easily to
n dimensions. For our purpose, however, 3 dimensions would suffice.) A vector
represented by
vector is given by
is
in Cartesian (rectangular) coordinates. The magnitude of the
A unit vector in any direction has a magnitude (length) 1. The unit vectors parallel to the
cartesian
and
coordinates are usually designated by
these unit vectors, the vector
and
respectively. In terms of
is written
2

Any vector in 3 dimension may be written in this fashion. The vectors
are said to form a
basis. In fact, any three non-collinear vectors may be used as a basis. The basis vectors used here
are perpendicular to one another. A unit vector along the direction of
is
Vector Addition
Sum of two vectors
and
is a third vector. If
then
Geometrically, the vector addition is represented by parallelogram law or the triangle law,
illustrated below.
Scalar Multiplication
The effect of multiplying a vector by a real number is to multiply its magnitude by without
a change in direction (except where is negative, in which case the vector gets inverted). In the
component representation, each component gets multiplied by the scalar
Scalar multiplication is distributive in addition, i.e.
Two vectors may be multiplied to give either a scalar or a vector.
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Scalar Product (The Dot products)
The dot product of two vectors
and
is a scalar given by the product of the magnitudes of
the vectors times the cosine of the angle (
) between the two
In terms of the components of the vectors
Note that
Dot product is commutative and distributive
Two vectors are orthogonal if
Dot products of the cartesian basis vectors are as follows
Exercise 1
Show that the vectors
and
are orthogonal.
Vector Product (The Cross Product)
The cross product of two vectors
is a vector whose magnitude is
where is the angle between the two vectors. The direction of the product vector is
,
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