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

6 Dynamics of Mechanical Systems

Two kinds of vectors occur so frequently that they deserve special attention: zero vectors
and unit vectors. A zero vector is simply a vector with magnitude zero. A unit vector is a
vector with magnitude one; unit vectors have no dimensions or units.
Zero vectors are useful in equations involving vectors. Unit vectors are useful for
separating the characteristics of vectors. That is, every vector V may be expressed as the
product of a scalar and a unit vector. In such a product, the scalar represents the magnitude
of the vector and the unit vector represents the direction. Specifically, if V is written as:

V = s n (1.5.1)

where s is a scalar and n is a unit vector, then s and n are:

s = V and n = V V (1.5.2)
/
This means that given any non-zero vector V we can always find a unit vector n with the
same direction as V; thus, n represents the direction of V.

1.6 Reference Frames and Coordinate Systems

We can represent a reference frame by identifying it with a coordinate–axes system such
as a Cartesian coordinate system. Specifically, we have three mutually perpendicular lines,
called axes, which intersect at a point O called the origin, as in Figure 1.6.1. The space is
then filled with “points” that are located relative to O by distances from O to P measured
along lines parallel to the axes. These distances form sets of three numbers, called the
coordinates of the points. Each point is then associated with its coordinates.
The points in space may also be located relative to O by introducing additional lines
conveniently associated with the points together with the angles these lines make with
the mutually perpendicular axes. The coordinates of the points may then involve these
angles.
To illustrate these concepts, consider first the Cartesian coordinate system shown in
Figure 1.6.2, where the axes are called X, Y, and Z. Let P be a typical point in space. Then
the coordinates of P are the distances x, y, and z from P to the planes Y–Z, Z–X, and X–Y,
respectively.

Z
P(x,y,z)

O
O
Y

X
FIGURE 1.6.1 FIGURE 1.6.2
A reference frame with origin O. Cartesian coordinate system.