Practice Exercises

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3 Imagine a circle centered at the origin in polar coordinates The equation for the circle is r = a, where a is a real-number constant What other equation, if any, represents the same circle 4 In Fig 3-5 on page 43, suppose that each radial increment is p units What s the value of the constant a in this case What s the equation of the pair of spirals (Here are a couple of reminders: The radial increments are the concentric circles The value of a in this situation turns out negative) 5 Figure 3-8 shows a line L and a circle C in polar coordinates Line L passes through the origin, and every point on L is equidistant from the horizontal and vertical axes Circle C is centered at the origin Each radial division represents 1 unit What s the polar equation representing L when we restrict the angles to positive values smaller than 2p What s the polar equation representing C (Here s a hint: Both equations can be represented in two ways) 6 When we examine Fig 3-8, we can see that L and C intersect at two points P and Q What are the polar coordinates of P and Q, based on the information given in Problem 5 (Here s a hint: Both points can be represented in two ways)

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Intersection point P

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p /2

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Line L

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Circle C

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3p /2

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Intersection point Q

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Figure 3-8 Illustration for Problems 5 through 10

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Each radial division is 1 unit

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Polar Two-Space

7 Solve the system of equations from the solution to Problem 5, verifying the polar coordinates of points P and Q in Fig 3-8 8 Based on the information given in Problem 5, what are the Cartesian xy-coordinate equations of line L and circle C in Fig 3-8 9 Solve the system of equations from the solution to Problem 8 to determine the Cartesian coordinates of the intersection points P and Q in Fig 3-8 10 Based on the polar coordinates of points P and Q in Fig 3-8 (the solutions to Problems 6 and 7), use the conversion formulas to derive the Cartesian coordinates of those two points

CHAPTER

Vector Basics

We can define the length of a line segment that connects two points, but the direction is ambiguous If we want to take the direction into account, we must make a line segment into a vector Mathematicians write vector names as bold letters of the alphabet Alternatively, a vector name can be denoted as a letter with a line or arrow over it

The Cartesian Way

In diagrams and graphs, a vector is drawn as a directed line segment whose direction is portrayed by putting an arrow at one end When working in two-space, we can describe vectors in Cartesian coordinates or in polar coordinates Let s look at the Cartesian way first

Endpoints, locations, and notations Figure 4-1 shows four vectors drawn on a Cartesian coordinate grid Each vector has a beginning (the originating point) and an end space (the terminating point) In this situation, any of the four vectors can be defined according to two independent quantities:

The length (magnitude) The way it points (direction) It doesn t matter where the originating or terminating points actually are The important thing is how the two points are located with respect to each other Once a vector has been defined as having a specific magnitude and direction, we can slide it around all over the coordinate plane without changing its essential nature We can always think of the originating point for a vector as being located at the coordinate origin (0,0) When we place a vector so that its originating point is at (0,0), we say that the vector is in standard form The standard form is convenient in Cartesian