Page 24 - Dynamics of Mechanical Systems

0593_C01_fm Page 5 Monday, May 6, 2002 1:43 PM

Introduction 5

1.5 Vector Review
Because vectors are used extensively in the text, 2B
it is helpful to review a few of their fundamental A B
concepts. We will expand this review in Chapter
2. Mathematically, a vector may be defined as an
element of a vector space (see, for example, Ref-
erences 1.1 to 1.3). For our purposes, we may think
of a vector simply as a directed line segment.
Figure 1.5.1 shows some examples of vectors
as directed line segments. In this context, vectors
are seen to have several characteristics: magni- -A
tude, orientation, and sense. The magnitude of a
FIGURE 1.5.1
vector is simply its length; hence, in a graphical Vectors depicted as directed line segments.
representation as in Figure 1.5.1, the magnitude
is simply the geometrical length. (Observe, for example, that vector 2B has a length and
magnitude twice that of vector B.) The orientation of a vector refers to its inclination in
space; this inclination is usually measured relative to some fixed coordinate system. The
sense of a vector is designated by the position of the arrowhead in the geometrical repre-
sentation. Observe, for example, in Figure 1.5.1 that vectors A and –A have opposite sense.
The combined characteristics of orientation and sense are sometimes called the direction
of a vector.
In this book, we will use vectors to represent forces, velocities, and accelerations. We
will also use them to locate points and to indicate directions. The units of a vector are
those of its magnitude. In turn, the units of the magnitude depend upon the quantity the
vector represents. For example, if a vector represents a force, its magnitude is measured
in force units such as Newtons (N) or pounds (lb). Alternatively, if a vector represents
velocity, its units might be meters per second (m/sec) or feet per second (ft/sec). Hence,
vectors representing different quantities will have graphical representations with different
length scales. (A review of specific systems of units is presented in Section 1.7.)
Because vectors have the characteristics of magnitude and direction they are distinct
from scalars, which are simply elements of a real or complex number system. For example,
the magnitude of a vector is a scalar; the direction of a vector is not a scalar. To distinguish
vectors from scalars, vectors are printed in bold-face type, such as V. Also, because the
magnitude of a vector is never negative (length is
never negative), absolute-value signs are used to 2V
designate the magnitude, such as V. V (1/2)V
In the next chapter, we will review algebraic oper- -(3/2)V
ations of vectors, such as the addition and multipli-
cation of vectors. In preparation for this, it is helpful
to review the concept of multiplication of vectors by
scalars. Specifically, if a vector V is multiplied by a
scalar s, the product, written as sV, is a vector whose
magnitude is sV, where s is the absolute
value of the scalar s. The direction of sV is the same
as that of V if s is positive and opposite that of V if FIGURE 1.5.2
s is negative. Figure 1.5.2 shows some examples of Examples of products of scalars and a
vector V.
products of scalars and vectors.