Environmental Engineering Reference
In-Depth Information
D
B
CD
AB
C
A
a
Figure 1.1 Displacement of a particle from point
A
to point
B
is illustrated by the directed
line segment
AB
. Parallel transport of this line gives the displacement from point
C
to
point
D
. The displacement vector
a
is not associated with any particular starting point.
an operation that is known as parallel transport. Now the displacement is denoted
CD
but its direction and magnitude are the same. It should be clear that there is
an infinity of such displacements that may be obtained by parallel transport of the
directed line segment. The displacement vector
a
has the magnitude and direction
common to this infinite set of displacements but is not associated with a particular
position in space. This is an important point which sometimes causes confusion
since vectors are illustrated as directed line segments, which appear to have a
well defined beginning and an end in space: A vector has magnitude and direction
but not location. The position of a particle in space may be given generally by a
position vector
r
only in conjunction with a fixed point of origin.
Now, all of this assumes that we understand what it means for lines to be
parallel. At this point we assume that we are working in Euclidean space, which
means that parallel lines remain equidistant everywhere, i.e. they never intersect.
In non-Euclidean spaces, such as the two-dimensional surface of a sphere, parallel
lines do intersect
3
and extra mathematics is required to specify how local geometries
are transported to different locations in the space. For the moment, since we have
no need of non-Euclidean geometry, we will rest our discussion of vectors firmly
on the familiar Euclidean notion of parallel lines. Later, when we consider the
space-time geometry associated with relativistic motion we will be forced to drop
this deep-rooted assumption about the nature of space.
So far, we have been considering only vectors that are associated with displace-
ments from one point to another. Their utility is far more wide ranging than that
though: vectors are used to represent other interesting quantities in physics. For
example the electric field strength in the vicinity of an electric charge is correctly
represented by specifying both its magnitude and direction, i.e. it is a vector. Since
it is important to maintain the distinction between vectors and ordinary numbers
3
For example lines of longitude meet at the poles.
