Unit+6

Chapter 6: Vectors, Parametric Equations, and Polar Equations
The topics in this chapter have been moved from Chapters 8 and 9 in the previous edition to form a single chapter that might be described as "other ways to coordinatize the plane." Not all students will need this material in a first-year calculus course, but some will (for example, high school students intending to take AP Calculus BC). Physics students will use the material on vectors and on planar motion defined parametrically. Not all classes will cover this chapter. Although all the topics are beautiful mathematics, instructors that arrive at this point pressed for time may need to start making some choices about which "extra" topics to throw into a minimal precalculus course: vectors or infinite series? polar form or statistics? DeMoivre's Theorem or a preview of calculus? In some cases, state or departmental curriculum guides will force these decisions. We tried to make this textbook as lean and lively as possible, but we did not omit any "precalculus" topic if a good argument could be made for leaving it in. Your teacher must now face the unfortunate reality that this policy might have resulted in a course that can not be teachable in the time allotted. (We hope that this is less true of this book than it is of many others.) If your class skips this chapter and you find the material interesting, why not study it on your own anyway?

Section 6.1 Vectors in the Plane
**Objectives**

You will be able to apply the algebra of vectors and use vectors to solve real-world problems. **Key Ideas**
 * Component form of a vector || Scalar multiplication ||
 * Directed line segment || Speed ||
 * Direction angle || Standard position of a vector ||
 * Equal vectors || Standard unit vectors **i** and **j** ||
 * Equivalent line segments || Terminal point ||
 * Horizontal component || Unit vector ||
 * Initial point || Vector addition ||
 * Length or magnitude (of a vector) || Velocity ||
 * Linear combination (of vectors) || Vertical component ||
 * Study Tips** We define vectors in two ways, both of which have their uses: as an equivalence class of directed line segments (useful geometrically) and as an ordered pair of components (useful algebraically). We use angled brackets (e.g., [[image:http://media.pearsoncmg.com/aw/aw_demana_precalc_8/cw/images/ch6/eq1.gif width="25" height="17" align="absMiddle"]] to represent vectors, but you need be warned that many books (perhaps even their physics book) use parentheses, relying on context to distinguish them from coordinates of points in the plane. Also, we use bold type to distinguish vector variables (e.g., **v**) to represent vector variables, while many other books use arrows (e.g., [[image:http://media.pearsoncmg.com/aw/aw_demana_precalc_8/cw/images/ch6/eq2.gif width="9" height="12" align="absMiddle"]]).

Section 6.2 Dot Product of Vectors
**Objectives**

You will be able to calculate dot products, the angle between two vectors, and projections of vectors. You will be able to apply these concepts to real-world problems involving components of force and work. **Key Ideas** **Study Tips** Studying dot products will help you understand matrix multiplication. What is hard to understand in this section is why the dot product (also called the **inner product**) is defined in such a strange way. The text mentions the applications to components of force and work, to which we add the observation that linear equations can be written as vector equations using the dot product.
 * Angle between two vectors || Newton-meter ||
 * Dot product || Orthogonal vectors ||
 * Foot-pound || Vector projection ||
 * Joule || Work ||



Section 6.3 Parametric Equations and Motion
**Objectives**

You will be able to graphs curves parametrically and solve application problems using parametric equations. **Key Ideas**
 * Parameter || Parametric curve ||
 * Parameter interval || Parametric equations ||

Section 6.4 Polar Coordinates
**Objectives**

You will be able to convert planar coordinates and curves from rectangular to polar coordinates and vice-versa. **Key Ideas**
 * Polar coordinate system || Directed distance ||
 * Directed angle || Pole ||



Section 6.5 Graphs of Polar Equations
**Objectives**

You will be able to graph and analyze planar curves described in polar coordinates. **Key Ideas**
 * Polar graph || Limaon curves ||
 * Cardioid || Rose curves ||
 * Lemniscate curves || Spiral of Archimedes ||

Section 6.6 DeMoivre's Theorem and //n//th Roots
**Objectives**

You will be able to represent complex numbers in polar form and use that representation to simplify certain algebraic operations, especially the finding of //n//th roots. **Key Ideas**
 * Argument (of a complex number) || //n//th root of unity ||
 * DeMoivre's Theorem || Polar form (of a complex number) ||
 * Modulus (of a complex number) || Trigonometric form (of a complex number) ||