3.1
Coordinate Systems
59
e often need to work with physical quantities that have both numerical and
directional properties. As noted in Section 2.1, quantities of this nature are
represented by vectors. This chapter is primarily concerned with vector alge-
bra and with some general properties of vector quantities. We discuss the addition
and subtraction of vector quantities, together with some common applications to
physical situations.
Vector quantities are used throughout this text, and it is therefore imperative
that you master both their graphical and their algebraic properties.
COORDINATE SYSTEMS
Many aspects of physics deal in some form or other with locations in space. In
Chapter 2, for example, we saw that the mathematical description of an object’s
motion requires a method for describing the object’s position at various times.
This description is accomplished with the use of coordinates, and in Chapter 2 we
used the cartesian coordinate system, in which horizontal and vertical axes inter-
sect at a point taken to be the origin (Fig. 3.1). Cartesian coordinates are also
called
rectangular coordinates.
Sometimes it is more convenient to represent a point in a plane by its
plane po-
lar coordinates
(
r
,
±
), as shown in Figure 3.2a. In this
polar coordinate system, r
is the
distance from the origin to the point having cartesian coordinates (
x
,
y
), and
is
the angle between
r
and a ±xed axis. This ±xed axis is usually the positive
x
axis,
and
is usually measured counterclockwise from it. From the right triangle in Fig-
ure 3.2b, we ±nd that sin
²
y
/
r
and that cos
²
x
/
r
. (A review of trigonometric
functions is given in Appendix B.4.) Therefore, starting with the plane polar coor-
dinates of any point, we can obtain the cartesian coordinates, using the equations
(3.1)
(3.2)
Furthermore, the de±nitions of trigonometry tell us that
(3.3)
(3.4)
These four expressions relating the coordinates (
x
,
y
) to the coordinates (
r
,
)
apply only when
is de±ned, as shown in Figure 3.2a—in other words, when posi-
tive
is an angle measured
counterclockwise
from the positive
x
axis. (Some scienti±c
calculators perform conversions between cartesian and polar coordinates based on
these standard conventions.) If the reference axis for the polar angle
is chosen
to be one other than the positive
x
axis or if the sense of increasing
is chosen dif-
ferently, then the expressions relating the two sets of coordinates will change.