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03-Vectors

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

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c h a p t e r 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 C h a p t e r O u t l i n e 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 defined pat- tern, 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 located relative to the hive? (E. Webber/Visuals Unlimited) 58 P U Z Z L E R P U Z Z L E R

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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 fixed axis. This fixed axis is usually the positive x axis, and is usually measured counterclockwise from it. From the right triangle in Fig- ure 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 the plane polar coor- dinates of any point, we can obtain the cartesian coordinates, using the equations (3.1) (3.2) Furthermore, the definitions 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 defined, as shown in Figure 3.2a—in other words, when posi- tive 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 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.
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