I
n our study of physics, we 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 vector quantities. This chapter is primarily concerned with vector algebra 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.
3.1
Coordinate Systems
Many aspects of physics involve a description of a location in space. In Chapter 2,
for example, we saw that the mathematical description of an object’s motion re-
quires a method for describing the object’s position at various times. This descrip-
tion is accomplished with the use of coordinates, and in Chapter 2 we used the
Cartesian coordinate system, in which horizontal and vertical axes intersect at a
point deﬁned as the origin (Fig. 3.1). Cartesian coordinates are also called
rectangu-
lar coordinates
.
Sometimes it is more convenient to represent a point in a plane by its
plane polar co-
ordinates
(
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 be-
tween a line drawn from the origin to the point and a ﬁxed axis. This ﬁxed axis is usu-
ally the positive
x
axis, and
is usually measured counterclockwise from it. From the
right triangle in Figure 3.2b, we ﬁnd 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 coordinates of any point, we can obtain the Cartesian coordinates by using the
equations
x
²
r
cos
(3.1)
y
²
r
sin
(3.2)
59
(–3, 4)
y
O
Q
P
(
x
,
y
)
(5, 3)
x
Figure 3.1
Designation of points in a Cartesian coor-
dinate system. Every point is labeled with coordinates
(
x
,
y
).
O
(
x, y
)
y
x
r
θ
(a)
(b)
x
r
y
sin
θ
=
y
r
cos
θ
=
x
r
tan
θ
=
x
y
Active Figure 3.2
(a) The plane
polar coordinates of a point are
represented by the distance
r
and
the angle
, where
is measured
counterclockwise from the positive
x
axis. (b) The right triangle used
to relate (
x
,
y
) to (
r
,
).