1. Section 9.1 Polar Coordinates

2. x OriginPole Polar axis

3. r O Pole

4. The Polar Plane Coordinates (r, θ)

5. Polar axis O Pole 4

7. 4 O 7

8. 8

9. Section 9.2 Polar Equations and Graphs 9

10. Identify and graph the equation: r = 2 Circle with center at the pole and radius 2.

13. Let a be a nonzero real number, the graph of the equation is a horizontal line y = a

14. Let a be a nonzero real number, the graph of the equation is a vertical line x = a

15. 15

16. 16

17. Let a be a positive or negative real number. Then, Circle: radius a ; center at (0, a) Circle: radius a ; center at (a, 0). 17

18. Symmetry with Respect to the Polar Axis (x-axis): 18

19. Symmetry with Respect to the Line (y-axis)

20. Symmetry with Respect to the Pole (Origin):

21. Tests for Symmetry Symmetry with Respect to the Polar Axis (x-axis): Replace θ by - θ Symmetry with Respect to the Line (y-axis): Replace θ by Π - θ Symmetry with Respect to the Pole (Origin): Replace r by -r If an equivalent equation results then the graph is symmetric with respect to the given pole or line. 21

22. Specific Types of Polar Graphs Cardioids (heart shaped) where a > 0. The graph passes through the pole. Limacons without an inner loop (French word for snail) where a > 0, b > 0, and a > b. The graph does not pass through the pole. 22

23. Limacons with an inner loop (French word for snail) where a > 0, b > 0, and a < b. The graph passes through the pole twice. Rose Curves If n is even and not zero, the graph has 2n petals. If n is odd and not one or negative one, the graph has n petals.

24. Lemniscates (Greek word for propeller) where a is non-zero. The graph will be propeller shaped. 24

25. Section 9.3 The Complex Plane 25

26. Real Axis Imaginary Axis O The Complex Plane

27. Real Axis Imaginary Axis Ox y z is the magnitude of z = x + yi 27

28. Cartesian Form Polar Form AND 28

29. 4 -3 Real Axis Imaginary Axis 29

32. DeMoivre’s Theorem 32

35. 35

36. Section 9.4 Vectors 36

37. A vector is a quantity that has both magnitude and direction. Vectors in the plane can be represented by arrows. The length of the arrow represents the magnitude of the vector. The arrowhead indicates the direction of the vector.

38. P Q Initial Point Terminal Point

39. The vector v whose magnitude is 0 is called the zero vector, 0. if they have the same magnitude and direction. Two vectors v and w are equal, written

40. Initial point of v Terminal point of w Vector Addition

41. Vector addition is commutative. Vector addition is associative.

42. Properties of Scalar Products 42

43. Use the vectors illustrated below to graph each expression. 43

46. 46

47. 47

48. Let i be a unit vector along the pos. x-axis; Let j be a unit vector along the pos. y-axis. If v has initial point at the origin O and terminal point at P = (a, b), then 48

49. a P = (a, b) v = ai + bj b The scalars a and b are called components of the vector v = ai + bj.

50. Position Vector The position vector re-positions the vector so that the initial point is the origin. 50

52. O v = 5i + 3j 52

53. Section 9.6 Vectors in Space 53

54. In space, each point is associated with an ordered triple of real numbers. Through a fixed point, the origin, O, draw three mutually perpendicular lines, the x-axis, y- axis, and z-axis. z y x 2 -2 2 2 O

55. Distance Formula in Space

56. 56

57. If v is a vector with initial point at the origin O and terminal point at P = (a, b, c), then we can represent v in terms of the vectors i, j, and k as v = ai + bj + ck P = (a, b, c) v = ai + bj + ck Position Vector 59

59. Properties of Dot Product If u, v, and w are vectors, then Commutative Property Distributive Property 59

60. Section 9.7 The Cross Product

61. If v = a 1 i + b 1 j + c 1 k and w = a 2 i + b 2 j + c 2 k are two vectors in space, the cross product v x w is defined as the vector v x w = (b 1 c 2 - b 2 c 1 )i - (a 1 c 2 - a 2 c 1 )j + (a 1 b 2 - a 2 b 1 )k Example: If v = 3i + 2j + 4k and w = 2i + j + 2k, find the cross product v x w.

63. A 3 by 3 determinant is symbolized by

64. Evaluate the following determinant.

65. Determinates can be used to find cross products. Find v x w is v = 3i + 2j + 4k and w = 2i + j + 2k.

66. If u, v, and w are vectors in space and if a is a scalar, then u x u = 0 u x v = -(v x u) a(u x v) = (au) x v = u x (av) u x (v x w) = (u x v) + (u x w) Algebraic Properties of the Cross Product

67. If u, v, and w are vectors in space and if a is a scalar, then u x v is orthogonal to both u and v u x v=0 if and only if u and v are orthogonal Geometric Properties of the Cross Product

68. Find a vector that is orthogonal to u = 2i - 3j + k and v = -3i - j + k.