SECTION 6.7. The Dot Product. Preview Exercises. 754 Chapter 6 Additional Topics in Trigonometry. 7 w u 7 2 =?. 7 v 77w7

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1 754 Chapter 6 Additional Topics in Trigonometry 115. Yo ant to fly yor small plane de north, bt there is a 75-kilometer ind bloing from est to east. a. Find the direction angle for here yo shold head the plane if yor speed relatie to the grond is 310 kilometers per hor. b. If yo increase yor air speed, shold the direction angle in part (a) increase or decrease? Explain yor anser. Preie Exercises Exercises ill help yo prepare for the material coered in the next section Find the obtse angle, ronded to the nearest tenth of a degree, satisfying c os = 3(-1) + (-2)(4), here = 3i - 2j and = -i + 4j If = -2i + 6j, find the folloing ector: 2(-2) + 4(-6) Consider the triangle formed by ectors,, and. (0, 0) y (a 1, b 1 ) (a 2, b 2 ) a. Use the magnitdes of the three ectors to rite the La of Cosines for the triangle shon in the figre: =?. b. Use the coordinates of the points shon in the figre to rite algebraic expressions for 7 7, 7 7 2, 7 7, 7 7 2, 7 7, and x SECTION 6.7 The Dot Prodct Objecties Find the dot prodct of to ectors. Find the angle beteen to ectors. Use the dot prodct to determine if to ectors are orthogonal. Find the projection of a ector onto another ector. Express a ector as the sm of to orthogonal ectors. Compte ork. T alk abot hard ork! I can see the eightlifter s mscles qiering from the exertion of holding the barbell in a stationary position aboe her head. Still, I m not sre if she s doing as mch ork as I am, sitting at my desk ith my brain qiering from stdying trigonometric fnctions and their applications. Wold it srprise yo to kno that neither yo nor the eightlifter are doing any ork at all? The definition of ork in physics and mathematics is not the same as hat e mean by ork in eeryday se. To nderstand hat is inoled in real ork, e trn to a ne ector operation called the dot prodct.

2 Section 6.7 The Dot Prodct 755 Find the dot prodct of to ectors. The Dot Prodct of To Vectors The operations of ector addition and scalar mltiplication reslt in ectors. By contrast, the dot prodct of to ectors reslts in a scalar (a real nmber), rather than a ector. Definition of the Dot Prodct If = a 1 i + b 1 j and = a 2 i + b 2 j are ectors, the dot prodct # is defined as follos: # = a1 a 2 + b 1 b 2. The dot prodct of to ectors is the sm of the prodcts of their horizontal components and their ertical components. EXAMPLE 1 Finding Dot Prodcts If = 5i - 2j and = -3i + 4j, find each of the folloing dot prodcts: a. # b. # c. #. SOLUTION To find each dot prodct, mltiply the to horizontal components, and then mltiply the to ertical components. Finally, add the to prodcts. a. =5( 3)+( 2)(4)= 15-8= 23 Mltiply the horizontal components and mltiply the ertical components of = 5i 2j and = 3i + 4j. b. = 3(5)+4( 2)= 15-8= 23 Mltiply the horizontal components and mltiply the ertical components of = 3i + 4j and = 5i 2j. c. =5(5)+( 2)( 2)=25+4=29 Mltiply the horizontal components and mltiply the ertical components of = 5i 2j and = 5i 2j. Check Point 1 If = 7i - 4j and = 2i - j, find each of the folloing dot prodcts: a. # b. # c. #. In Example 1 and Check Point 1, did yo notice that # and # prodced the same scalar? The fact that # = # follos from the definition of the dot prodct. Properties of the dot prodct are gien in the folloing box. Proofs for some of these properties are gien in the appendix. Properties of the Dot Prodct If,, and are ectors, and c is a scalar, then 1. # = # 2. # ( + ) = # + # 3. 0 # = 0 4. # = ( c) # = c( # ) = # (c)

3 756 Chapter 6 Additional Topics in Trigonometry y The Angle beteen To Vectors (0, 0) (a 1, b 1 ) (a 2, b 2 ) x The La of Cosines can be sed to derie another formla for the dot prodct. This formla ill gie s a ay to find the angle beteen to ectors. Figre 6.64 shos ectors = a 1 i + b 1 j and = a 2 i + b 2 j. By the definition of the dot prodct, e kno that # = a1 a 2 + b 1 b 2. Or ne formla for the dot prodct inoles the angle beteen the ectors, shon as in the figre. Apply the La of Cosines to the triangle shon in the figre. FIGURE = cos Use the La of Cosines. = (a 1 a 2 )i + (b 1 b 2 )j = (a 1 a 2 ) 2 + (b 1 b 2 ) 2 = a 1 i + b 1 j = a b 1 2 = a 2 i + b 2 j = a b 2 2 ( a 1 - a 2 ) 2 + (b 1 - b 2 ) 2 = (a b 1 2 ) + (a b 2 2 ) cos Sbstitte the sqares of the magnitdes of ectors,, and into the La of Cosines. a 1 2-2a 1 a 2 + a b 1 2-2b 1 b 2 + b 2 2 = a b a b cos Sqare the binomials sing (A - B) 2 = A 2-2AB + B 2. -2a 1 a 2-2b 1 b 2 = cos Sbtract a 1 2, a 2 2, b 1 2, and b 2 2 from both sides of the eqation. a 1 a 2 +b 1 b 2 = cos Diide both sides by -2. By definition, = a 1 a 2 + b 1 b 2. # = cos Sbstitte # for the expression on the left side of the eqation. Alternatie Formla for the Dot Prodct If and are to nonzero ectors and is the smallest nonnegatie angle beteen them, then # = cos. Find the angle beteen to ectors. Soling the formla in the box for cos gies s a formla for finding the angle beteen to ectors: Formla for the Angle beteen To Vectors If and are to nonzero ectors and is the smallest nonnegatie angle beteen and, then cos = # and = cos -1 # EXAMPLE 2 Finding the Angle beteen To Vectors Find the angle beteen the ectors = 3i - 2j and = -i + 4j, shon in Figre 6.65 at the top of the next page. Rond to the nearest tenth of a degree.

4 ( 1, 4) = i + 4j 5 4 y (3, 2) 4 = 3i 2j 5 FIGURE 6.65 Finding the angle beteen to ectors x SOLUTION Use the formla for the angle beteen to ectors. co s = = # (3i - 2j) # (-i + 4j) (-2) 2 2(-1) (-1) + (-2)(4) = = The angle beteen the ectors is Section 6.7 The Dot Prodct 757 Begin ith the formla for the cosine of the angle beteen to ectors. Sbstitte the gien ectors in the nmerator. Find the magnitde of each ector in the denominator. Find the dot prodct in the nmerator. Simplify in the denominator. Perform the indicated operations. = cos Use a calclator Check Point 2 Find the angle beteen the ectors = 4i - 3j and = i + 2j. Rond to the nearest tenth of a degree. Use the dot prodct to determine if to ectors are orthogonal. FIGURE 6.67 Orthogonal ectors: = 90 and cos = 0 y = i + 2j = 6i 3j 5 FIGURE 6.68 Orthogonal ectors x Parallel and Orthogonal Vectors To ectors are parallel hen the angle beteen the ectors is 0 or 180. If = 0, the ectors point in the same direction. If = 180, the ectors point in opposite directions. Figre 6.66 shos parallel ectors. = 0 and cos = 1. = 180 and cos = -1. Vectors Vectors point in the point in opposite directions. same direction. FIGURE 6.66 Parallel ectors To ectors are orthogonal hen the angle beteen the ectors is 90, shon in Figre (The ord orthogonal, rather than perpendiclar, is sed to describe ectors that meet at right angles.) We kno that # = 7 777cos. If and are orthogonal, then # = 7 777cos 90 = (0) = 0. Conersely, if and are ectors sch that # = 0, then 7 7 = 0 or 7 7 = 0 or cos = 0. If co s = 0, then = 90, so and are orthogonal. The preceding discssion is smmarized as follos: The Dot Prodct and Orthogonal Vectors To nonzero ectors and are orthogonal if and only if # = 0. Becase 0 # = 0, the zero ector is orthogonal to eery ector. EXAMPLE 3 Determining Whether Vectors Are Orthogonal Are the ectors = 6i - 3j and = i + 2j orthogonal? SOLUTION The ectors are orthogonal if their dot prodct is 0. Begin by finding #. # = (6i - 3j) # (i + 2j) = 6(1) + (-3)(2) = 6-6 = 0 The dot prodct is 0. Ths, the gien ectors are orthogonal. They are shon in Figre Check Point 3 Are the ectors = 2i + 3j and = 6i - 4j orthogonal?

5 758 Chapter 6 Additional Topics in Trigonometry Find the projection of a ector onto another ector. F 1 F 2 F FIGURE 6.69 Projection of a Vector onto Another Vector Yo kno ho to add to ectors to obtain a resltant ector. We no reerse this process by expressing a ector as the sm of to orthogonal ectors. By doing this, yo can determine ho mch force is applied in a particlar direction. For example, Figre 6.69 shos a boat on a tilted ramp. The force de to graity, F, is plling straight don on the boat. Part of this force, F 1, is pshing the boat don the ramp. Another part of this force, F 2, is pressing the boat against the ramp, at a right angle to the incline. These to orthogonal ectors, F 1 and F 2, are called the ector components of F. Notice that F = F 1 + F 2. A method for finding F 1 and F 2 inoles projecting a ector onto another ector. Figre 6.70 shos to nonzero ectors, and, ith the same initial point. The angle beteen the ectors,, is acte in Figre 6.70(a) and obtse in Figre 6.70(b). A third ector, called the ector projection of onto, is also shon in each figre, denoted by proj. proj proj FIGURE 6.70(a) FIGURE 6.70(b) Ho is the ector projection of onto formed? Dra the line segment from the terminal point of that forms a right angle ith a line throgh, shon in red. The projection of onto lies on a line throgh, and is parallel to ector. This ector begins at the common initial point of and. It ends at the point here the dashed red line segment intersects the line throgh. Or goal is to determine an expression for proj. We begin ith its magnitde. By the definition of the cosine fnction, proj cos = This is the magnitde of the ector projection of onto. 7 7 cos = 7 proj 7 Mltiply both sides by proj 7 = 7 7 cos. Reerse the to sides. We can rerite the right side of this eqation and obtain another expression for the magnitde of the ector projection of onto. To do so, se the alternate formla for the dot prodct, # = cos. Diide both sides of # = cos by 7 7 : # 7 7 = 7 7 cos. The expression on the right side of this eqation, 7 7 cos, is the same expression that appears in the formla for 7 proj 7. Ths, 7 proj 7 = 7 7 cos = # 7 7.

6 Section 6.7 The Dot Prodct 759 We se the formla for the magnitde of proj to find the ector itself. This is done by finding the scalar prodct of the magnitde and the nit ector in the direction of. proj = a b = a b 2 This is the magnitde of the ector projection of onto. This is the nit ector in the direction of. The Vector Projection of onto If and are to nonzero ectors, the ector projection of onto is proj = # EXAMPLE 4 Finding the Vector Projection of One Vector onto Another If = 2i + 4j and = -2i + 6j, find the ector projection of onto. proj y x SOLUTION The ector projection of onto is fond sing the formla for proj. proj = # = (2i + 4j) # (-2i + 6j) 1 2(-2) (-2) + 4(6) = = = 1 2 (-2i + 6j) = -i + 3j The three ectors,,, and proj, are shon in Figre FIGURE 6.71 The ector projection of onto Express a ector as the sm of to orthogonal ectors. Check Point 4 If = 2i - 5j and = i - j, find the ector projection of onto. We se the ector projection of onto, proj, to express as the sm of to orthogonal ectors. The Vector Components of Let and be to nonzero ectors. Vector can be expressed as the sm of to orthogonal ectors, 1 and 2, here 1 is parallel to and 2 is orthogonal to. 1 = proj = # 7 7 2, 2 = - 1 Ths, = The ectors 1 and 2 are called the ector components of. The process of expressing as is called the decomposition of into 1 and 2. EXAMPLE 5 Decomposing a Vector into To Orthogonal Vectors Let = 2i + 4j and = -2i + 6j. Decompose into to ectors, 1 and 2, here 1 is parallel to and 2 is orthogonal to.

7 760 Chapter 6 Additional Topics in Trigonometry SOLUTION The ectors = 2i + 4j and = -2i + 6j are the ectors e orked ith in Example 4. We se the formlas in the box on the preceding page. 1 = proj = -i + 3j We obtained this ector in Example 4. 2 = - 1 = (2i + 4j) - (-i + 3j) = 3i + j Check Point 5 Let = 2i - 5j and = i - j. (These are the ectors from Check Point 4.) Decompose into to ectors, 1 and 2, here 1 is parallel to and 2 is orthogonal to. Compte ork. Work: An Application of the Dot Prodct The bad nes: Yor car jst died. The good nes: It died on a leel road jst 200 feet from a gas station. Exerting a constant force of 90 ponds, and not necessarily histling as yo ork, yo manage to psh the car to the gas station. Force: 90 ponds A B 200 feet Althogh yo did not histle, yo certainly did ork pshing the car 200 feet from point A to point B. Ho mch ork did yo do? If a constant force F is applied to an object, moing it from point A to point B in the direction of the force, the ork, W, done is W = (magnitde of force)(distance from A to B). Yo pshed ith a force of 90 ponds for a distance of 200 feet. The ork done by yor force is W = (90 ponds)(200 feet) or 18,000 foot-ponds. Work is often measred in foot-ponds or in neton-meters. The photo on the left shos an adlt plling a small child in a agon. Work is being done. Hoeer, the sitation is not qite the same as pshing yor car. Pshing the car, the force yo applied as along the line of motion. By contrast, the force of the adlt plling the agon is not applied along the line of the agon s motion. In this case, the dot prodct is sed to determine the ork done by the force. Definition of Work The ork, W, done by a force F moing an object from A to B is W = F # AB ". When compting ork, it is often easier to se the alternatie formla for the dot prodct. Ths, W=F AB= F AB cos. F is the magnitde of the force. AB is the distance oer hich the constant force is applied. is the angle beteen the force and the direction of motion. It is correct to refer to W as either the ork done or the ork done by the force.

8 Section 6.7 The Dot Prodct ponds 35 FIGURE 6.72 Compting ork done plling a sled 200 feet EXAMPLE 6 Compting Work A child plls a sled along leel grond by exerting a force of 30 ponds on a rope that makes an angle of 35 ith the horizontal. Ho mch ork is done plling the sled 200 feet? SOLUTION The sitation is illstrated in Figre The ork done is W= F AB cos =(30)(200) cos Magnitde of the force is 30 ponds. Distance is 200 feet. The angle beteen the force and the sled s motion is 35. Ths, the ork done is approximately 4915 foot-ponds. Check Point 6 A child plls a agon along leel grond by exerting a force of 20 ponds on a handle that makes an angle of 30 ith the horizontal. Ho mch ork is done plling the agon 150 feet? CONCEPT AND VOCABULARY CHECK Fill in each blank so that the reslting statement is tre. 1. If = a 1 i + b 1 j and = a 2 i + b 2 j are ectors, the prodct #, called the, is defined as # =. 2. If and are to nonzero ectors and is the smallest nonnegatie angle beteen them, then # =. 3. If # = 0, then the ectors and are. 4. Tre or false: Gien to nonzero ectors and, can be decomposed into to ectors, one parallel to and the other orthogonal to. 5. Tre or false: The definition of ork indicates that ork is a ector. EXERCISE SET 6.7 Practice Exercises In Exercises 1 8, se the gien ectors to find # and #. 1. = 3i + j, = i + 3j 2. = 3i + 3j, = i + 4j 3. = 5i - 4j, = -2i - j 4. = 7i - 2j, = -3i - j 5. = -6i - 5j, = -10i - 8j 6. = -8i - 3j, = -10i - 5j 7. = 5i, = j 8. = i, = -5j In Exercises 9 16, let = 2i - j, = 3i + j, and = i + 4j. Find each specified scalar. 9. # ( + ) 10. # ( + ) 11. # + # 12. # + # 13. ( 4 ) # 14. ( 5 ) # ( # ) ( # ) In Exercises 17 22, find the angle beteen and. Rond to the nearest tenth of a degree. 17. = 2i - j, = 3i + 4j 18. = -2i + 5j, = 3i + 6j 19. = -3i + 2j, = 4i - j 20. = i + 2j, = 4i - 3j 21. = 6i, = 5i + 4j 22. = 3j, = 4i + 5j In Exercises 23 32, se the dot prodct to determine hether and are orthogonal. 23. = i + j, = i - j 24. = i + j, = -i + j 25. = 2i + 8j, = 4i - j 26. = 8i - 4j, = -6i - 12j 27. = 2i - 2j, = -i + j 28. = 5i - 5j, = i - j 29. = 3i, = -4i 30. = 5i, = -6i 31. = 3i, = -4j 32. = 5i, = -6j

9 762 Chapter 6 Additional Topics in Trigonometry In Exercises 33 38, find proj. Then decompose into to ectors, 1 and 2, here 1 is parallel to and 2 is orthogonal to. 33. = 3i - 2j, = i - j 34. = 3i - 2j, = 2i + j 35. = i + 3j, = -2i + 5j 36. = 2i + 4j, = -3i + 6j 37. = i + 2j, = 3i + 6j 38. = 2i + j, = 6i + 3j Practice Pls In Exercises 39 42, let = -i + j, = 3i - 2j, and = -5j. Find each specified scalar or ector # (3-4) # (5-3) 41. proj ( + ) 42. proj ( - ) In Exercises 43 44, find the angle, in degrees, beteen and. 43. = 2 cos 4p 3 i + 2 sin 4p 3 j, = 3 cos 3p 2 i + 3 sin 3p 2 j 44. = 3 cos 5p 3 i + 3 sin 5p 3 j, = 2 cos pi + 2 sin pj In Exercises 45 50, determine hether and are parallel, orthogonal, or neither. 45. = 3i - 5j, = 6i - 10j 46. = -2i + 3j, = -6i + 9j 47. = 3i - 5j, = 6i + 10j 48. = -2i + 3j, = -6i - 9j 49. = 3i - 5j, = 6i j 50. = -2i + 3j, = -6i - 4j Application Exercises 51. The components of = 240i + 300j represent the respectie nmber of gallons of reglar and premim gas sold at a station. The components of = 2.90i j represent the respectie prices per gallon for each kind of gas. Find # and describe hat the anser means in practical terms. 52. The components of = 180i + 450j represent the respectie nmber of one-day and three-day ideos rented from a ideo store. The components of = 3i + 2j represent the prices to rent the one-day and three-day ideos, respectiely. Find # and describe hat the anser means in practical terms. 53. Find the ork done in pshing a car along a leel road from point A to point B, 80 feet from A, hile exerting a constant force of 95 ponds. Rond to the nearest foot-pond. 54. Find the ork done hen a crane lifts a 6000-pond bolder throgh a ertical distance of 12 feet. Rond to the nearest foot-pond. 55. A agon is plled along leel grond by exerting a force of 40 ponds on a handle that makes an angle of 32 ith the horizontal. Ho mch ork is done plling the agon 100 feet? Rond to the nearest foot-pond. 56. A agon is plled along leel grond by exerting a force of 25 ponds on a handle that makes an angle of 38 ith the horizontal. Ho mch ork is done plling the agon 100 feet? Rond to the nearest foot-pond. 57. A force of 60 ponds on a rope is sed to pll a box p a ramp inclined at 12 from the horizontal. The figre shos that the rope forms an angle of 38 ith the horizontal. Ho mch ork is done plling the box 20 feet along the ramp? 60 ponds 58. A force of 80 ponds on a rope is sed to pll a box p a ramp inclined at 10 from the horizontal. The rope forms an angle of 33 ith the horizontal. Ho mch ork is done plling the box 25 feet along the ramp? 59. A force is gien by the ector F = 3i + 2j. The force moes an object along a straight line from the point (4, 9) to the point (10, 20). Find the ork done if the distance is measred in feet and the force is measred in ponds. 60. A force is gien by the ector F = 5i + 7j. The force moes an object along a straight line from the point (8, 11) to the point (18, 20). Find the ork done if the distance is measred in meters and the force is measred in netons. 61. A force of 4 ponds acts in the direction of 50 to the horizontal. The force moes an object along a straight line from the point (3, 7) to the point (8, 10), ith distance measred in feet. Find the ork done by the force. 62. A force of 6 ponds acts in the direction of 40 to the horizontal. The force moes an object along a straight line from the point (5, 9) to the point (8, 20), ith the distance measred in feet. Find the ork done by the force. 63. Refer to Figre 6.69 on page 758. Sppose a boat eighs 700 ponds and is on a ramp inclined at 30. Represent the force de to graity, F, sing F = -700j. a. Write a nit ector along the ramp in the pard direction. b. Find the ector projection of F onto the nit ector from part (a). c. What is the magnitde of the ector projection in part (b)? What does this represent? 64. Refer to Figre 6.69 on page 758. Sppose a boat eighs 650 ponds and is on a ramp inclined at 30. Represent the force de to graity, F, sing F = -650j. a. Write a nit ector along the ramp in the pard direction. b. Find the ector projection of F onto the nit ector from part (a). c. What is the magnitde of the ector projection in part (b)? What does this represent? 38 12

10 Section 6.7 The Dot Prodct 763 Writing in Mathematics 65. Explain ho to find the dot prodct of to ectors. 66. Using ords and no symbols, describe ho to find the dot prodct of to ectors ith the alternatie formla # = cos. 67. Describe ho to find the angle beteen to ectors. 68. What are parallel ectors? 69. What are orthogonal ectors? 70. Ho do yo determine if to ectors are orthogonal? 71. Dra to ectors, and, ith the same initial point. Sho the ector projection of onto in yor diagram. Then describe ho yo identified this ector. 72. Ho do yo determine the ork done by a force F in moing an object from A to B hen the direction of the force is not along the line of motion? 73. A eightlifter is holding a barbell perfectly still aboe his head, his body shaking from the effort. Ho mch ork is the eightlifter doing? Explain yor anser. 74. Describe one ay in hich the eeryday se of the ord ork is different from the definition of ork gien in this section. Critical Thinking Exercises Make Sense? In Exercises 75 78, determine hether each statement makes sense or does not make sense, and explain yor reasoning. 75. Althogh I expected ector operations to prodce another ector, the dot prodct of to ectors is not a ector, bt a real nmber. 76. I e noticed that heneer the dot prodct is negatie, the angle beteen the to ectors is obtse. 77. I m orking ith a nit ector, so its dot prodct ith itself mst be The eightlifter does more ork in raising 300 kilograms aboe her head than Atlas, ho is spporting the entire orld. In Exercises 79 81, se the ectors = a 1 i + b 1 j, = a 2 i + b 2 j, and = a 3 i + b 3 j, to proe the gien property. 79. # = # 80. ( c) # = c( # ) 81. # ( + ) = # + # 82. If = -2i + 5j, find a ector orthogonal to. 83. Find a ale of b so that 15i - 3j and -4i + bj are orthogonal. 84. Proe that the projection of onto i is ( # i)i. 85. Find to ectors and sch that the projection of onto is. Grop Exercise 86. Grop members shold research and present a report on nsal and interesting applications of ectors. Preie Exercises Exercises ill help yo prepare for the material coered in the first section of the next chapter. 87. a. Does (4, -1) satisfy x + 2y = 2? b. Does (4, -1) satisfy x - 2y = 6? 88. Graph x + 2y = 2 and x - 2y = 6 in the same rectanglar coordinate system. At hat point do the graphs intersect? 89. Sole: 5(2x - 3) - 4x = 9.

11 764 Chapter 6 Additional Topics in Trigonometry CHAPTER 6 Smmary, Reie, and Test SUMMARY DEFINITIONS AND CONCEPTS 6.1 and 6.2 The La of Sines; The La of Cosines a. The La of Sines a sin A = b sin B = c sin C b. The La of Sines is sed to sole SAA, ASA, and SSA (the ambigos case) triangles. The ambigos case may reslt in no triangle, one triangle, or to triangles; see the box on page 685. c. The area of a triangle eqals one-half the prodct of the lengths of to sides times the sine of their inclded angle. d. The La of Cosines a 2 = b 2 + c 2-2bc cos A b 2 = a 2 + c 2-2ac cos B c 2 = a 2 + b 2-2ab cos C e. The La of Cosines is sed to find the side opposite the gien angle in an SAS triangle; see the box at the bottom of the page on page 695. The La of Cosines is also sed to find the angle opposite the longest side in an SSS triangle; see the box on page 696. f. Heron s Formla for the Area of a Triangle The area of a triangle ith sides a, b, and c is 2s(s - a)(s - b)(s - c), here s is one-half its perimeter: s = 1 2 (a + b + c). 6.3 and 6.4 Polar Coordinates; Graphs of Polar Eqations a. A point P in the polar coordinate system is represented by (r, ), here r is the directed distance from the pole to the point and is the angle from the polar axis to line segment OP. The elements of the ordered pair ( r, ) are called the polar coordinates of P. See Figre 6.20 on page 703. When r in ( r, ) is negatie, a point is located r nits along the ray opposite the terminal side of. Important information abot the sign of r and the location of the point (r, ) is fond in the box on page 703. b. Mltiple Representations of Points If n is any integer, (r, ) = (r, + 2np) or (r, ) = (-r, + p + 2np). c. Relations beteen Polar and Rectanglar Coordinates EXAMPLES Ex. 1, p. 683 ; Ex. 2, p. 684 ; Ex. 3, p. 685 ; Ex. 4, p. 686 Ex. 5, p. 687 Ex. 6, p. 688 Ex. 1, p. 696 ; Ex. 2, p. 697 Ex. 4, p. 698 Ex. 1, p. 703 Ex. 2, p. 704 x = r cos, y = r sin, x 2 + y 2 = r 2, tan = y x d. To conert a point from polar coordinates (r, ) to rectanglar coordinates (x, y), se x = r cos and y = r sin. e. A point in rectanglar coordinates (x, y) can be conerted to polar coordinates (r, ). Use the procedre in the box on page 707. f. To conert a rectanglar eqation to a polar eqation, replace x ith r cos and y ith r sin. g. To conert a polar eqation to a rectanglar eqation, se one or more of Ex. 3, p. 706 Ex. 4, p. 707 ; Ex. 5, p. 708 Ex. 6, p. 708 Ex. 7, p. 709 r 2 = x 2 + y 2, r cos = x, r sin = y, and tan = y x. It is often necessary to do something to the gien polar eqation before sing the preceding expressions. h. A polar eqation is an eqation hose ariables are r and. The graph of a polar eqation is the set of all points hose polar coordinates satisfy the eqation. i. Polar eqations can be graphed sing point plotting and symmetry (see the box on page 716 ). Ex. 1, p. 714 Ex. 2, p. 716

12 Section Smmary, 6.7 Reie, The Dot and Prodct Test 765 DEFINITIONS AND CONCEPTS j. The graphs of r = a cos and r = a sin are circles. See the box on page 715. The graphs of r = a { b sin and r = a { b cos are called limaçons ( a 7 0 and b 7 0 ), shon in the box on page 718. The graphs of r = a sin n and r = a cos n, a 0, are rose cres ith 2n petals if n is een and n petals if n is odd. See the box on page 720. The graphs of r 2 = a 2 sin 2 and r 2 = a 2 cos 2, a 0, are called lemniscates and are shon in the box on page Complex Nmbers in Polar Form; DeMoire s Theorem a. The complex nmber z = a + bi is represented as a point (a, b) in the complex plane, shon in Figre 6.38 on page 726. b. The absolte ale of z = a + bi is z = a + bi = 2a 2 + b 2. c. The polar form of z = a + bi is z = r(cos + i sin ), here a = r cos, b = r sin, r = 2a 2 + b 2, and tan = b. We call r the modls and the argment of z, ith 0 6 2p. a d. Mltiplying Complex Nmbers in Polar Form: Mltiply modli and add argments. See the box on page 729. e. Diiding Complex Nmbers in Polar Form: Diide modli and sbtract argments. See the box on page 730. f. DeMoire s Theorem is sed to find poers of complex nmbers in polar form. [ r(cos + i sin )] n = r n (cos n + i sin n) g. DeMoire s Theorem can be sed to find roots of complex nmbers in polar form. The n distinct nth roots of r(cos + i sin ) are or 1 n rjcosa + 2pk n 1 n rjcosa k n b + i sina + 2pk b R n b + i sina k b R, n EXAMPLES Ex. 3, p. 717 ; Ex. 4, p. 719 ; Ex. 5, p. 720 Ex. 1, p. 727 Ex. 2, p. 727 Ex. 3, p. 728 ; Ex. 4, p. 729 Ex. 5, p. 730 Ex. 6, p. 731 Ex. 7, p. 732 ; Ex. 8, p. 732 Ex. 9, p. 733 ; Ex. 10, p. 734 here k = 0, 1, 2, c, n Vectors a. A ector is a directed line segment. b. Eqal ectors hae the same magnitde and the same direction. c. The ector k, the scalar mltiple of the ector and the scalar k, has magnitde k 7 7. The direction of k is the same as that of if k 7 0 and opposite if k 6 0. d. The sm +, called the resltant ector, can be expressed geometrically. Position and so that the terminal point of coincides ith the initial point of. The ector + extends from the initial point of to the terminal point of. e. The difference of to ectors, -, is defined as + (-). f. The ector i is the nit ector hose direction is along the positie x@axis. The ector j is the nit ector hose direction is along the positie y@a x is. g. Vector, from (0, 0) to (a, b), called a position ector, is represented as = ai + bj, here a is the horizontal component and b is the ertical component. The magnitde of is gien by 7 7 = 2a 2 + b 2. h. Vector from (x 1, y 1 ) to ( x 2, y 2 ) is eqal to the position ector = (x 2 - x 1 )i + (y 2 - y 1 )j. In rectanglar coordinates, the term ector refers to the position ector in terms of i and j that is eqal to it. i. Operations ith Vectors in Terms of i and j If = a 1 i + b 1 j and = a 2 i + b 2 j, then + = (a 1 + a 2 )i + (b 1 + b 2 )j - = (a 1 - a 2 )i + (b 1 - b 2 )j k = (ka 1 )i + (kb 1 )j j. The zero ector 0 is the ector hose magnitde is 0 and is assigned no direction. Many properties of ector addition and scalar mltiplication inole the zero ector. Some of these properties are listed in the box on page 746. Ex. 1, p. 740 Figre 6.52, p. 741 Figre 6.53, p. 741 Figre 6.54, p. 742 Ex. 2, p. 743 Ex. 3, p. 744 Ex. 4, p. 745 ; Ex. 5, p. 745 ; Ex. 6, p. 746

13 766 Chapter 6 Additional Topics in Trigonometry DEFINITIONS AND CONCEPTS k. The ector is the nit ector that has the same direction as. 7 7 l. A ector ith magnitde 7 7 and direction angle, the angle that makes ith the positie x@axis, can be expressed in terms of its magnitde and direction angle as 6.7 The Dot Prodct = 7 7 cos i sin j. a. Definition of the Dot Prodct If = a 1 i + b 1 j and = a 2 i + b 2 j, the dot prodct of and is defined by # = a1 a 2 + b 1 b 2. b. Alternatie Formla for the Dot Prodct: # = cos, here is the smallest nonnegatie angle beteen and c. Angle beteen To Vectors EXAMPLES Ex. 7, p. 747 Ex. 8, p. 748 ; Ex. 9, p. 748 Ex. 1, p. 755 Ex. 2, p. 756 c os = # and = cos -1 # d. To ectors are orthogonal hen the angle beteen them is 90. To sho that to ectors are orthogonal, sho that their dot prodct is zero. e. The ector projection of onto is gien by Ex. 3, p. 757 Ex. 4, p. 759 proj = # f. A ector may be expressed as the sm of to orthogonal ectors, called the ector components. See the box at the bottom of the page on page 759. g. The ork, W, done by a force F moing an object from A to B is W = F # h AB. h Ths, W = 7 F 77AB 7 cos, here is the angle beteen the force and the direction of motion. Ex. 5, p. 759 Ex. 6, p. 761 REVIEW EXERCISES 6.1 and 6.2 In Exercises 1 12, sole each triangle. Rond lengths to the nearest tenth and angle measres to the nearest degree. If no triangle exists, state no triangle. If to triangles exist, sole each triangle. 1. A = 70, B = 55, a = B = 107, C = 30, c = B = 66, a = 17, c = a = 117, b = 66, c = A = 35, B = 25, c = A = 39, a = 20, b = C = 50, a = 3, c = 1 8. A = 162, b = 11.2, c = a = 26.1, b = 40.2, c = A = 40, a = 6, b = B = 37, a = 12.4, b = A = 23, a = 54.3, b = In Exercises 13 16, find the area of the triangle haing the gien measrements. Rond to the nearest sqare nit. 13. C = 42, a = 4 feet, b = 6 feet 14. A = 22, b = 4 feet, c = 5 feet 15. a = 2 meters, b = 4 meters, c = 5 meters 16. a = 2 meters, b = 2 meters, c = 2 meters 17. The A-frame cabin shon belo is 35 feet ide. The roof of the cabin makes a 55 angle ith the cabin s base. Find the length of one side of the roof from its grond leel to the peak. Rond to the nearest tenth of a foot ft To cars leae a city at the same time and trael along straight highays that differ in direction by 80. One car aerages 60 miles per hor and the other aerages 50 miles per hor. Ho far apart ill the cars be after 30 mintes? Rond to the nearest tenth of a mile.

14 Section Smmary, 6.7 Reie, The Dot and Prodct Test To airplanes leae an airport at the same time on different rnays. One flies on a bearing of N66.5 W at 325 miles per hor. The other airplane flies on a bearing of S26.5 W at 300 miles per hor. Ho far apart ill the airplanes be after to hors? 20. The figre shos three roads that intersect to bond a trianglar piece of land. Find the lengths of the other to sides of the land to the nearest foot feet In Exercises 47 49, test for symmetry ith respect to a. the polar axis. b. the line = p 2. c. the pole. 47. r = cos 48. r = 3 sin 49. r 2 = 9 cos 2 In Exercises 50 56, graph each polar eqation. Be sre to test for symmetry. 50. r = 3 cos 51. r = sin 52. r = sin r = 2 + cos 54. r = sin 55. r = 1-2 cos 56. r 2 = cos A commercial piece of real estate is priced at $5.25 per sqare foot. Find the cost, to the nearest dollar, of a trianglar lot measring 260 feet by 320 feet by 450 feet. 6.3 and 6.4 In Exercises 22 27, plot each point in polar coordinates and find its rectanglar coordinates. 22. (4, 60 ) 23. (3, 150 ) 24. a-4, 4p 3 b 25. a-2, 5p 4 b 26. a-4, - p 2 b 27. a-2, - p 4 b In Exercises 28 30, plot each point in polar coordinates. Then find another representation (r, ) of this point in hich a. r 7 0, 2p 6 6 4p. b. r 6 0, p. c. r 7 0, -2p a3, p 6 b 29. a2, 2p 3 b 30. a3.5, p 2 b In Exercises 31 36, the rectanglar coordinates of a point are gien. Find polar coordinates of each point. 31. ( -4, 4 ) 32. ( 3, -3 ) 33. (5, 12) 34. ( -3, 4 ) 35. ( 0, -5 ) 36. (1, 0) In Exercises 37 39, conert each rectanglar eqation to a polar eqation that expresses r in terms of x + 3y = x 2 + y 2 = ( x - 6) 2 + y 2 = 3 6 In Exercises 40 46, conert each polar eqation to a rectanglar eqation. Then se yor knoledge of the rectanglar eqation to graph the polar eqation in a polar coordinate system. 40. r = = 3p r cos = r = 5 csc 44. r = 3 cos r cos + r sin = r 2 sin 2 = In Exercises 57 60, plot each complex nmber. Then rite the complex nmber in polar form. Yo may express the argment in degrees or radians i i i i In Exercises 61 64, rite each complex nmber in rectanglar form. If necessary, rond to the nearest tenth ( c os i sin 60 ) ( c os i sin 210 ) acos 2p 3 + i sin 2p 3 b (cos i sin 100 ) In Exercises 65 67, find the prodct of the complex nmbers. Leae ansers in polar form. 65. z 1 = 3(cos 40 + i sin 40 ) z 2 = 5(cos 70 + i sin 70 ) 66. z 1 = cos i sin 210 z 2 = cos 55 + i sin z 1 = 4acos 3p 7 + i sin 3p 7 b z 2 = 10acos 4p 7 + i sin 4p 7 b In Exercises 68 70, find the qotient z 1 z 2 of the complex nmbers. Leae ansers in polar form. 68. z 1 = 10(cos 10 + i sin 10 ) z 2 = 5(cos 5 + i sin 5 ) 69. z 1 = 5acos 4p 3 + i sin 4p 3 b z 2 = 10acos p 3 + i sin p 3 b 70. z 1 = 2acos 5p 3 + i sin 5p 3 b z 2 = cos p 2 + i sin p 2

15 768 Chapter 6 Additional Topics in Trigonometry In Exercises 71 75, se DeMoire s Theorem to find the indicated poer of the complex nmber. Write ansers in rectanglar form. 71. [ 2 ( c os i sin 20 )] [ 4 ( c os i sin 50 )] J 1 2 acos p 14 + i sin p 14 b R i ( -2-2i) 5 In Exercises 76 77, find all the complex roots. Write roots in polar form ith in degrees. 76. The complex sqare roots of 49(cos 50 + i sin 50 ) 77. The complex cbe roots of 125(cos i sin 165 ) In Exercises 78 81, find all the complex roots. Write roots in rectanglar form. 78. The complex forth roots of 16acos 2p 3 + i sin 2p 3 b 79. The complex cbe roots of 8i 80. The complex cbe roots of The complex fifth roots of -1 - i 6.6 In Exercises 82 84, sketch each ector as a position ector and find its magnitde. 82. = -3i - 4j 83. = 5i - 2j 84. = -3j In Exercises 85 86, let be the ector from initial point P 1 to terminal point P 2. Write in terms of i and j. 85. P 1 = (2, -1), P 2 = (5, -3 ) 86. P 1 = (-3, 0), P 2 = (-2,-2 ) In Exercises 87 90, let = i - 5j and = -2i + 7j. Find each specified ector or scalar In Exercises 91 92, find the nit ector that has the same direction as the ector. 91. = 8i - 6j 92. = -i + 2j 93. The magnitde and direction angle of are 7 7 = 12 and = 60. Express in terms of i and j. 94. The magnitde and direction of to forces acting on an object are 100 ponds, N25 E, and 200 ponds, N80 E, respectiely. Find the magnitde, to the nearest pond, and the direction angle, to the nearest tenth of a degree, of the resltant force. 95. Yor boat is moing at a speed of 15 miles per hor at an angle of 25 pstream on a rier floing at 4 miles per hor. The sitation is illstrated in the figre belo. 6.7 y 25 Boat s speed: 15 miles per hor Boat s elocity relatie to the grond x Water s speed: 4 miles per hor a. Find the ector representing yor boat s elocity relatie to the grond. b. What is the speed of yor boat, to the nearest mile per hor, relatie to the grond? c. What is the boat s direction angle, to the nearest tenth of a degree, relatie to the grond? 96. If = 5i + 2j, = i - j, and = 3i - 7j, find # ( + ). In Exercises 97 99, find the dot prodct #. Then find the angle beteen and to the nearest tenth of a degree. 97. = 2i + 3j, = 7i - 4j 98. = 2i + 4j, = 6i - 11j 99. = 2i + j, = i - j In Exercises , se the dot prodct to determine hether and are orthogonal = 12i - 8j, = 2i + 3j 101. = i + 3j, = -3i - j In Exercises , find proj. Then decompose into to ectors, 1 and 2, here 1 is parallel to and 2 is orthogonal to = -2i + 5j, = 5i + 4j 103. = -i + 2j, = 3i - j 104. A heay crate is dragged 50 feet along a leel floor. Find the ork done if a force of 30 ponds at an angle of 42 is sed.

16 Section Smmary, 6.7 Reie, The Dot and Prodct Test 769 CHAPTER 6 TEST 1. In obliqe triangle ABC, A = 34, B = 68, and a = 4.8. Find b to the nearest tenth. 2. In obliqe triangle ABC, C = 68, a = 5, and b = 6. Find c to the nearest tenth. 3. In obliqe triangle ABC, a = 17 inches, b = 45 inches, and c = 32 inches. Find the area of the triangle to the nearest sqare inch. 4. Plot a4, 5p b in the polar coordinate system. Then rite to 4 other ordered pairs (r, ) that name this point. 5. If the rectanglar coordinates of a point are (1, -1), find polar coordinates of the point. 6. Conert x 2 + (y + 8) 2 = 64 to a polar eqation that expresses r in terms of. 7. Conert to a rectanglar eqation and then graph: r = -4 sec. In Exercises 8 9, graph each polar eqation. 8. r = 1 + sin 9. r = cos 10. W r it e i in polar form. In Exercises 11 13, perform the indicated operation. Leae ansers in polar form ( c os i sin 15 ) # 10(cos 5 + i sin 5 ) 12. 2acos p 2 + i sin p 2 b 4acos p 3 + i sin p 3 b 13. [ 2 ( c os i sin 10 )] Find the three cbe roots of 27. Write roots in rectanglar form. 15. If P 1 = (-2, 3), P 2 = (-1, 5), and is the ector from P 1 to P 2, a. Write in terms of i and j. b. Find 7 7. In Exercises 16 19, let = -5i + 2j and = 2i - 4j. Find the specified ector, scalar, or angle # 18. the angle beteen and, to the nearest degree 19. pr oj 20. A small fire is sighted from ranger stations A and B. Station B is 1.6 miles de east of station A. The bearing of the fire from station A is N35 E and the bearing of the fire from station B is N50 W. Ho far, to the nearest tenth of a mile, is the fire from station A? 21. The magnitde and direction of to forces acting on an object are 250 ponds, N60 E, and 150 ponds, S45 E. Find the magnitde, to the nearest pond, and the direction angle, to the nearest tenth of a degree, of the resltant force. 22. A child is plling a agon ith a force of 40 ponds. Ho mch ork is done in moing the agon 60 feet if the handle makes an angle of 35 ith the grond? Rond to the nearest foot-pond. CUMULATIVE REVIEW EXERCISES (CHAPTERS P 6) Sole each eqation or ineqality in Exercises x 4 - x 3 - x 2 - x - 2 = sin 2-3 sin + 1 = 0, 0 6 2p 3. x 2 + 2x s in cos = - 1 2, 0 6 2p In Exercises 5 6, graph one complete cycle. 5. y = 3 sin(2x - p) 6. y = -4 cos px In Exercises 7 8, erify each identity. 7. s in csc - cos 2 = sin 2 8. c os a + 3p 2 b = sin 9. Find the slope and y@intercept of the line hose eqation is 2 x + 4y - 8 = 0. In Exercises 10 11, find the exact ale of each expression s in p 3-3 tan p s in 1tan In Exercises 12 13, find the domain of the fnction hose eqation is gien. 12. f(x) = 25 - x 13. g(x) = x - 3 x A ball is thron ertically pard from a height of 8 feet ith an initial elocity of 48 feet per second. The ball s height, s(t), in feet, after t seconds is gien by s(t) = -16t t + 8. After ho many seconds does the ball reach its maximm height? What is the maximm height?

17 770 Chapter 6 Additional Topics in Trigonometry 15. An object moes in simple harmonic motion described by d = 4 sin 5t, here t is measred in seconds and d in meters. Find a. the maximm displacement; b. the freqency; and c. the time reqired for one cycle. 16. Use a half-angle formla to find the exact ale of cos If = 2i + 7j and = i - 2j, find a. 3 - and b. #. 18. Express as a single logarithm ith a coefficient of 1: 1 2 log b x - log b (x ). 19. Write the slope-intercept form of the line passing throgh ( 4, -1) and (-8, 5 ). 20. Psychologists can measre the amont learned, L, at time t sing the model L = A(1 - e -kt ). The ariable A represents the total amont to be learned and k is the learning rate. A stdent preparing for the SAT has 300 ne ocablary ords to learn: A = 300. This particlar stdent can learn 20 ocablary ords after 5 mintes: If t = 5, L = 20. a. Find k, the learning rate, correct to three decimal places. b. Approximately ho many ords ill the stdent hae learned after 20 mintes? c. Ho long ill it take for the stdent to learn 260 ords?