Environmental Engineering Reference
In-Depth Information
P
z
axis
z
r
x
q
O
f
y
x
axis
Figure 1.6 Two 3-dimensional co-ordinate systems covering the same space. The Cartesian
co-ordinates consist of the set
(x,y,z)
. The spherical polar co-ordinates consist of the set
(r,θ,φ)
.
θ
and the azimuthal angle
φ
. The Cartesian and the spherical polar systems are
just two possible ways of mapping the same space, and it should be clear that
for any given physical problem there will be an infinite number of equally-valid
co-ordinate systems. The decision as to which one to use is based on the nature
of the problem, and the ease or difficulty of the calculation that results from the
choice.
Choosing a co-ordinate system immediately gives us a way to represent vectors.
Associated with any co-ordinate system are a set of unit vectors known as basis
vectors. Each co-ordinate has an associated basis vector that points in the direction
in which that co-ordinate is increasing. For example, in the 3D Cartesian system
i
points in the direction of increasing
x
, i.e. along the
x
-axis, while
j
and
k
point
along the
y
axes, respectively. Suppose that the position of a particle
relative to the origin is given by the vector
r
, known as the 'position vector' of the
particle. Then
r
can be written in terms of the Cartesian basis vectors as
−
and
z
−
r
=
x
i
+
y
j
+
z
k
,
(1.4)
where the numbers
(x,y,z)
are the Cartesian co-ordinates of the particle. The
magnitude of the position vector, which is the distance between the particle and
the origin, can be calculated by Pythagoras' Theorem and is
x
2
=
√
r
r
·
r
=
+
y
2
+
z
2
.
(1.5)
We have focussed upon a position vector in the Cartesian basis but we could
have talked about a force, or an acceleration or a magnetic field etc. Any vector
A
can be expressed in terms of its components
(A
x
,A
y
,A
z
)
according to
A
=
A
x
i
+
A
y
j
+
A
z
k
.
(1.6)
It is not our aim here to present a full discussion of the algebraic properties of
vectors. Some key results, which will prove useful later are listed in Table 1.1.
Very often, the motion of an object may be constrained to a known plane, such
as in the case of a ball on a pool table, or a planet in orbit around the Sun. In such
situations the full 3D co-ordinate system is not required and a two-dimensional
