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3 - Vectors

# 3 - Vectors - Chapter 3 Vectors CHAPTE R OUTLI N E 3.1...

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Chapter 3 Vectors CHAPTE R OUTLI N E 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 58 These controls in the cockpit of a commercial aircraft assist the pilot in maintaining control over the velocity of the aircraft—how fast it is traveling and in what direction it is traveling—allowing it to land safely. Quantities that are defined by both a magnitude and a di- rection, such as velocity, are called vector quantities. (Mark Wagner/Getty Images)

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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 y / r and that cos 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 r cos (3.1) y 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) θ (b) x r y sin θ = y r cos θ = x r tan θ = x y θ θ θ 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 , ).
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3 - Vectors - Chapter 3 Vectors CHAPTE R OUTLI N E 3.1...

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