03-Vectors - P U Z Z L E R When this honeybee gets back to...

chapter
Vectors
3.1 Coordinate Systems
3.2 Vector and Scalar Quantities
3.3 Some Properties of Vectors
3.4 Components of a Vector and Unit
Vectors
Chapter Outline
When this honeybee gets back to its
hive, it will tell the other bees how to re-
turn to the food it has found. By moving
in a special, very precisely deFned pat-
tern, the bee conveys to other workers
the information they need to Fnd a flower
bed. Bees communicate by “speaking in
vectors.” What does the bee have to tell
the other bees in order to specify where
the flower bed is located relative to the
hive?
(E. Webber/Visuals Unlimited)
58
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UZZLER
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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.