676 CHAPTER 9 Polar Coordinates; Vectors Section You should be able to . . . Example(s) Review Exercises 9.2 1 Identify and graph polar equations by converting to rectangular equations (p. 611) 1–3, 5–7 7(b)–10(b) 2 Graph polar equations using a graphing utility (p. 612) 4–7 11–13 3 Test polar equations for symmetry (p. 615) 8–11 11–13 4 Graph polar equations by plotting points (p. 616) 8–14 11–13 9.3 1 Plot points in the complex plane (p. 627) 1 16–18 2 Convert a complex number between rectangular form and polar form or exponential form (p. 628) 2, 3 14–18 3 Find products and quotients of complex numbers (p. 630) 4 19–21 4 Use De Moivre’s Theorem (p. 631) 5, 6 22–25 5 Find complex roots (p. 632) 7 26 9.4 1 Graph vectors (p. 639) 1 27, 28 2 Find a position vector (p. 640) 2 29, 30 3 Add and subtract vectors algebraically (p. 641) 3 31 4 Find a scalar multiple and the magnitude of a vector (p. 642) 4 29, 30, 32–34 5 Find a unit vector (p. 642) 5 35 6 Find a vector from its direction and magnitude (p. 643) 6 36, 37 7 Model with vectors (p. 644) 8–10 59, 60 9.5 1 Find the dot product of two vectors (p. 651) 1 46, 47 2 Find the angle between two vectors (p. 652) 2 46, 47 3 Determine whether two vectors are parallel (p. 653) 3 50–52 4 Determine whether two vectors are orthogonal (p. 654) 4 50–52 5 Decompose a vector into two orthogonal vectors (p. 654) 5, 6 53, 54, 62 6 Compute work (p. 656) 7 61 9.6 1 Find the distance between two points in space (p. 660) 1 38 2 Find position vectors in space (p. 661) 2 39 3 Perform operations on vectors (p. 662) 3–5 40–42 4 Find the dot product (p. 663) 6 48, 49 5 Find the angle between two vectors (p. 664) 7 48, 49 6 Find the direction angles of a vector (p. 664) 8–10 55 9.7 1 Find the cross product of two vectors (p. 669) 1–3 43, 44 2 Know algebraic properties of the cross product (p. 670) p. 671 57, 58 3 Know geometric properties of the cross product (p. 671) p. 672 56 4 Find a vector orthogonal to two given vectors (p. 672) 4 45 5 Find the area of a parallelogram (p. 672) 5 56 Review Exercises In Problems 1–3, plot each point given in polar coordinates, and find its rectangular coordinates. 1. π ( ) 3, 6 2. π ( ) −2, 4 3 3. π ( ) − − 3, 2 In Problems 4–6, the rectangular coordinates of a point are given. Find two pairs of polar coordinates θ ( ) r, for each point, one with > r 0 and the other with < r 0. Express θ in radians. 4. ( ) −3, 3 5. ( ) − 0, 2 6. ( ) 3, 4 In Problems 7–10, the variables r and θ represent polar coordinates. (a) Write each polar equation as an equation in rectangular coordinates ( ) x y , . (b) Identify the equation and graph it. 7. θ = r 2 sin 8. = r 5 9. θ π = 4 10. θ θ + − = r r r 4 sin 8 cos 5 2
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