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P U Z Z L E R
When this honeybee gets back to its hive, it will tell the other bees how to return to the food it has found. By moving in a special, very precisely defined pattern, the bee conveys to other workers the information they need to find 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 locatedrelative to the hive? (E. Webber/Visuals Unlimited)

c h a p t e r

Vectors

Chapter Outline
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

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3.1

Coordinate Systems

59

W
3.1

e often need to work with physical quantities that have both numerical and directional properties. As noted inSection 2.1, quantities of this nature are represented by vectors. 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 boththeir graphical and their algebraic properties.
Q (–3, 4)

y (x, y)

P

(5, 3) x

COORDINATE SYSTEMS

O

2.2

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 accomplishedwith the use of coordinates, and in Chapter 2 we used the cartesian coordinate system, in which horizontal and vertical axes intersect 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 polar coordinates (r, ), as shown in Figure 3.2a. In this polar coordinatesystem, r is the distance from the origin to the point having cartesian coordinates (x, y), and is the angle between r and a fixed axis. This fixed axis is usually the positive x axis, and is usually measured counterclockwise from it. From the right triangle in Figure 3.2b, we find that sin y/r and that cos x/r. (A review of trigonometric functions is given in Appendix B.4.) Therefore, starting with theplane polar coordinates of any point, we can obtain the cartesian coordinates, using the equations x y r cos r sin (3.1) (3.2)

Figure 3.1 Designation of points in a cartesian coordinate system. Every point is labeled with coordinates (x, y).
y (x, y) r

θ
O (a) x

Furthermore, the definitions of trigonometry tell us that tan r y x (3.3) y2 (3.4)

y sin θ = r cos θ = x θ r tan θ = y x r y√x 2

These four expressions relating the coordinates (x, y) to the coordinates (r, ) apply only when is defined, as shown in Figure 3.2a — in other words, when positive is an angle measured counterclockwise from the positive x axis. (Some scientific calculators perform conversions between cartesian and polar coordinates based on these standard conventions.) If the reference axis for the polarangle is chosen to be one other than the positive x axis or if the sense of increasing is chosen differently, then the expressions relating the two sets of coordinates will change.

θ
x (b)

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, ).
You may want to read Talking Apes and Dancing Bees (1997) by Betsy Wyckoff.

Quick Quiz 3.1
Would the honeybee at the beginning of the chapter use cartesian or polar coordinates when specifying the location of the flower? Why? What is the honeybee using as an origin of coordinates?

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CHAPTER 3

Vectors

EXAMPLE 3.1

Polar Coordinates

The cartesian coordinates...