Bsc physics notes

  Vectors and Their Representations

A vector is an abstract quantity introduced by mathematicians and engineers to represent
a quantity which has magnitude and direction. Typically in this course we are interested in three
dimensional vectors representing position, velocity, and acceleration. A vector is generally
represented in its abstract form by an arrow whose length is proportional to the magnitude of the
vector quantity. We can deal directly with the vector quantity when doing certain operations such
as addition, subtraction, vector, and scalar multiplication by using graphical techniques and by
just indicating them, . However, to deal with calculations it is generally
necessary to use the REPRESENTATION of the vector. These representations usually require
one to define a coordinate system. In general the same vector can be represented in many
different ways, depending upon the coordinate system selected. We can pick different types of
coordinate systems, e.g. rectangular, spherical, plane polar, torroidal, elliptical, etc. and we can
pick different orientations of the same type of system. In either case, the representation of the
same vector will appear quite different. Although in general we are usually interested in different
orientations of the same type, at the present time we are interested in different types, in this case
rectangular and plane polar coordinate systems.
We generally define a coordinate system by a set of mutually orthogonal unit vectors,
called basis vectors. These vectors are of unit length and are perpendicular to each other forming
a unit triad. Typically the are designated by the symbol , where I indicates a direction. For
example the following is an equivalent representation of the generic vector in a rectangular
coordinate system:
or
(2)
Here, the Ai terms are called components of the vector and the
are the basis vectors for this coordinate system.
The position vector in a rectangular coordinate system is generally represented as
2

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(4)
with being the mutually orthogonal unit vectors along the x, y, and z axes respectively. The
values x, y, and z are the scalar components of the position vector .
All coordinate systems have two items in common, a reference plane, and a direction in
that reference plane. For rectangular coordinates one can think of the reference plane as the x-y
plane and the reference direction as the x direction. Further, the coordiante z is measured
perpendicular to the reference plane, giving us the coordinates (x, y, z). If we consider plane
polar (or cylindrical) coordinates, the reference plane is the one in which the radial component is
measured, (r), and the reference direction, the one from which the angle to the radial component
is measured (2). In addition, in cylindrical coordinates, the coordinate z is measured
perpendicular to the reference plane, giving us the coordinates (r, 2, z). In spherical coordinates
we can think of some equatorial-like plane as the reference plane. The magnitude of the position
vector (r) is one coordinate. The reference direction is that direction from which the angle to the
projection of the position vector on the reference plane is measured (2), and the elevation of the
position vector with respect to the reference plane is the third coordinate (N), giving us the
coordinates (r, 2, N).
Here, for reasons to become clear later, we are interested in plane polar (or cylindrical)
coordinates and spherical coordinates. Cylindrical coordinates have mutually orthogonal unit
vectors in the radial (parallel to the radius vector), transverse (perpendicular to the radius vector
in the plane of interest) and normal (perpendicular to the plane of interest). They are designated
as respectively. A generic vector would be represented as:
where are the scalar radial, transverse, and z components of the vector .
Spherical coordinates also have mutually orthogonal unit vectors in the radial (parallel to
the position vector), the longitudinal (parallel to the reference plane and perpendicular to the
position vector), and the elevation or latitude (along a constant longitude line and perpendicular
to the position and longitudinal unit vectors). A generic vector would be represented as:
(5)
where are the scalar radial, longitudinal, and latitudinal components of the vector .
It should be clear that the scalar components of the representation of the vector in plane polar
or spherical coordinates are not the same as those in rectangular coordinates. Hence the same
vector has a different representation in different types of coordinate systems. Also it should be
clear that the same vector will have a different representation in two rectangular coordinate
systems oriented in different directions.
3
(6)
Although the two representations are different in the two systems, they are related to each
other. If we consider the same vector represented in a rectangular coordinate system and in a
plane polar coordinate system, we have the following relations between the two representations:
where even the components Ar and A2 are different in the two different representations.
The relationship amongst the various components is called a transformation. We can
write the transformation matrix relating the cylindrical and spherical components of the vector to
the rectangular components. The results are for rectangular to cylindrical:
(7)
and for spherical:
(8)
Hence, although the representations of the same vector are different in different coordinate
systems, these representations are generally related to each other. Note that since these
transformation matrices are orthogonal matrices, their inverse can be obtained by just taking the
matrix transpose.
VECTOR ALGEBRA
Addition and Subtraction
Generally we can manipulate vector equations using the generic vector itself. However, it
is useful to know how to the basic vector algebra using the representations of the vector. For
4
(9)
(11)
(12)
addition and subtraction we have for the generic operation,
We can actually perform this operation in terms of the vector representations by noting that we
must have each vector represented in the same coordinate system. Then the addition and
subtraction operation is just the addition and subtraction of the vector components.
Scalar Product
The scalar (dot) product is generically given as where S is a scalar
(independent of coordinate system in which and are written). Furthermore, the definition is
. In terms of the vector representations, one can use the definition
of the scalar product to show it can be calculated simply as the sum of the products of like
components. Here we have
(10)
or
and the result is the same scalar, regardless of which representation is used.
The magnitude of a vector is given by:
Vector Product
The vector product is generically given as , where is a
unit vector perpendicular to the plane containing and directed in accordance with the
right hand rule. If you rotate into then the vector points in the direction in which a right
handed screw would advance. Alternatively, take your right hand, point you fingers in the
5
(13)
(14)
(16)
(17)
direction of and curl them towards and your thumb will be pointing in the direction of .
This same operation can be performed using the representations of the vector in the following
manner.
or
or
(15)
Vector Calculus
We generally have to deal with derivatives of the above vectors. Or particular interest
here is the representation in plane polar coordinates. It should be noted that the basis vectors in
the (inertial) rectangular coordinate system do not change in magnitude or direction, and hence
are constant. In plane polar coordinates, the basis vectors are constant in magnitude, but are
changing direction. Hence their derivatives are not zero. We can note the following development:
Similarly,
6
(19)
Then, since ,
(18)
In a similar manner we can represent the acceleration.
or
Similar, but much more complicated, calculations can be carried out for spherical
coordinates. The resulting unit vector rates can be determined to be:
(20)
Summary
The position, velocity, and acceleration for each coordinate system are given next.
Rectangular Coordinates Polar coordinates (in-plane components only)
(21)
7
(22)
Spherical Coordinates:
(24)
(25)
Finally we note:
and