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How To Study with DWFK Precalculus

Chapter 6: Vectors, Parametric Equations, and Polar Equations

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	Section 6.3	Section 6.5
Section 6.2	Section 6.4	Section 6.6

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Section 6.1 Vectors in the Plane

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., 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., ).

Technology Tips
Some graphing calculators will do vector addition and scalar multiplication. Others can be tricked into doing this vector algebra by doing matrix algebra for 1 x n matrices or by using lists:

The magnitude of a vector can also be found using lists, although not in a very user-friendly way:

It is, of course, hardly worth the effort. Sometimes technology can be less convenient.
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Section 6.2 Dot Product of Vectors

Key Ideas

Angle between two vectors	Newton-meter
Dot product	Orthogonal vectors
Foot-pound	Vector projection
Joule	Work

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. For example,

Technology Tips
Some graphing calculators will compute dot products. Others can be tricked into doing dot products by doing matrix multiplication, but one has to "transpose" the second matrix in the product to make the matrices conformable:

One really needs to ask if this is worth the effort. Students will want to know what the "T" means and why it is needed, and it will be hard to answer that question satisfactorily until matrices are encountered officially in Chapter 7.

Alternatively, here is the dot product using lists, still not very satisfying:

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Section 6.3 Parametric Equations and Motion

Key Ideas

Parameter	Parametric curve
Parameter interval	Parametric equations

Study Tips
This book introduced you to parametrically-defined functions in Section 1.4, but a full treatment of parametrics is introduced here for the first time. The primary application of parametrically-defined functions at this level (and, indeed, in calculus) is to problems of motion in two or more dimensions, of which we give several examples. The graphing calculator is used as a valuable tool for visualizing both the dynamic motion and the defined curve.

This section represents a radical departure from your parents' precalculus courses. Parametrically-defined functions can be comfortably studied at this level only because modern technology can graph them so easily. The early insights that this gives you into the modeling of motion problems will serve you well at every level of calculus.

Technology Tips
It is not only difficult to study this section without graphing calculators, but also highly inappropriate. You should be sure to follow the calculator-based examples and explorations as described in order to see this material at the level at which it is intended. (The section exercises provide plenty of supplementary examples for all. These are exercises that are intended to be done with calculators in one hand and a pencil in the other.)
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Section 6.4 Polar Coordinates

Key Ideas

Polar coordinate system	Directed distance
Directed angle	Pole

Study Tips
Many students will not study polar coordinates in their first year of calculus, but some will. (It is a topic in the course description for AP Calculus BC.) Polar coordinates provide a nice, spiraled review of trigonometry, and they open the door to some very interesting graphs in the next section. Thanks to graphing technology, you can produce the graphs whether you understand polar coordinates or not; consequently, you might have already seen these graphs and might have been wondering for years how to explain them.

Technology Tips
Some graphing calculators will convert polar coordinates to rectangular coordinates and vice-versa, but in this section it is best to do that converting on your own. (There are some exercises that specifically require calculators, but the elementary conversions lose their pedagogical value if you do not think through them using trigonometry.) The grapher becomes a more valuable tool in the next section.
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Section 6.5 Graphs of Polar Equations

Key Ideas

Polar graph	Lima�on curves
Cardioid	Rose curves
Lemniscate curves	Spiral of Archimedes

Study Tips
Classes that found it worthwhile to cover the previous section should definitely cover this one, as the graphs are the payoffs for learning the coordinate system. The curves are analyzed in the same manner as the graphs of the basic functions in Chapter 1.

It is important to observe that these graphs, although described by equations in the coordinates are graphed in the standard (x, y) coordinate plane. The coordinate axes should be labeled with y and x, as usual. (Note that the polar graph of in the (x, y) plane is a circle, while the graph of the same equation in an plane would be a sinusoid.)

Technology Tips
The fact that you can produce polar graphs quickly and accurately on your calculator is an asset to understanding and should be exploited. Once the graphs are produced, however, you should be able to analyze why they look that way. The examples and explorations of this section strike the appropriate balance between calculator and non-calculator activities, as do the exercises if the instructions are heeded.

The ability to produce accurate polar graphs freehand is a noble one artistically, but the mathematical understanding thereby gained (over analyzing them with technology) is dubious. In any event, it is not a goal of this section.
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Section 6.6 DeMoivre's Theorem and nth Roots

Key Ideas

Argument (of a complex number)	nth root of unity
DeMoivre's Theorem	Polar form (of a complex number)
Modulus (of a complex number)	Trigonometric form (of a complex number)

Study Tips
DeMoivre's Theorem is a lovely result and a fine review of trigonometric concepts, but realistically well out of the purview of calculus preparation. Teachers who are pressed for time should seriously consider whether it is appropriate to cover this section. (It is in the mandated course descriptions for some states.)

Be sure to stress, in keeping with one of the themes of this course, the connection that this theorem provides between the algebraic and geometric representations of complex numbers. For example, the roots of unity are evenly spaced around the unit circle. This should give students a quick geometric way to find all the complex solutions to equations like .

Technology Tips
Some calculators will convert rectangular coordinates to polar coordinates and vice-versa (polar coordinates being the same as trigonometric coordinates), although the polar form will probably be rather than . These two expressions look very different, but is an identity for all real numbers

Computer algebra systems (available on some calculators) will find complex roots for equations with real coefficients, enabling you to solve for nth roots of real numbers without DeMoivre's Theorem. Obviously, that use of the calculator works against the objectives of this section. Less sophisticated calculators will find a single root of a complex number when asked to compute a value for an expression like . Students who wish to find all roots to would still need to know how to use DeMoivre's Theorem to get the remaining roots from these primary roots. It is probably best to use calculators in the section exercises only to support answers (as we do in the examples).

Students who are good with trigonometry and good at calculator programming might try writing a program that will find all n nth roots of a complex number. The calculator will probably find one of them; getting the rest is harder than it looks.