1. What is meant by the magnitude of a vector and the direction of a vector? Æ are equal vectors. Æ and ST. 4. Vectors PQ 1 R

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1 3 Y IGT GUID I Da : pp s Da : pp s. 3 45, 48, 53 60, Qiz 3 s. 7 VRG Da : pp s Da : pp s. 3 48, 53 60, Qiz 3 s. 7 DVD Da : pp s Da : pp s O HDU pp s. 0 48, 53 60, Qiz 3 s. 7 XRI V eel : asier 0 eel : ore Difficlt 3 48 eel : ost Difficlt 49 5 HOOR H To ickl check stdent nderstanding of ke concepts, go oer the following eercises: s. 0, 6,, 8, 30, 3, 36, 44. ee also the Dail Homework Qiz: lackline aster (hapter 0 Resorce ook, p. ) Transparenc (p. 7) GUIDD RTI Vocablar heck oncept heck. The magnitde of a ector is the distance from its initial point to its terminal point, and the direction is the angle the ector makes with a horizontal line. kill heck. Æ = º, º ; Æ = 0, º3 ; Q Æ = 3, 3 ; UV Æ = 0, RTI D ITIO TUDT H tra ractice to help o master skills is on p. 80. TUDT H HOOR H ample : s. 0 0 ample : s. 4 ample 3: s. 5 9 ample 4: s ample 5: s hat is meant b the magnitde of a ector and the direction of a ector? In ercises 4, se the diagram. ee margin.. rite the component form of each ector. 3. Identif an parallel ectors. Æ, Q Æ Æ Æ ;, UV 4. Vectors Q Æ and T Æ are eal ectors. Æ lthogh T is not shown, the coordinates of its initial point are (º, º). Gie the coordinates of its terminal point. (, ) rite the ector in component form. ind the magnitde of the ector. Rond or answer to the nearest tenth , 5 ; 6.4 º4, º ; 4.5, º5 ; Use the ector in ercise 5. ind the direction of the ector relatie to east. abot 5 north of east 9. ind the sm of the ectors in ercises 5 and 6. 0, 3 IDIG GITUD rite the ector in component form. ind the magnitde of the ector. Rond or answer to the nearest tenth R œ 4, ; 4. º3, IDIG GITUD Draw ector Q Æ 6 ; 6.7 º4, º4 ; 5.7 in a coordinate plane. rite the component form of the ector and find its magnitde. Rond or answer to the nearest tenth. 3. (0, 0), Q(, 7), 7 ; (5, ), Q(, 6) º3, 5 ; (º3, ), Q(7, 6) 0, 4 ; (º4, º3), Q(, º7) 6, º4 ; (5, 0), Q(º, º4) º6, º4 ; (6, 3), Q(º, ) º8, º ; (º6, 0), Q(º5, º4), º4 ; (0, 5), Q(3, 5) 3, 0 ; 3 J V U œ 576 hapter 9 Right Triangles and Trigonometr 576

2 VIGTIO The gien ector represents the elocit of a ship at sea. ind the ship s speed, ronded to the nearest mile per hor. Then find the direction the ship is traeling relatie to the gien direction.. ind direction relatie to east. abot 6 mi/h; abot 9 north of east. ind direction relatie to east. abot 64 mi/h; abot 5 soth of east 0 T THIG TI XRI 6 hile direction is not important in finding magnitde, the component form depends on the direction. ake sre the terminal point is Q in this diagram. 3. ind direction relatie to west. 4. ind direction relatie to west. abot 57 mi/h; 45 north of west abot 64 mi/h; abot 39 soth of west 0 O In Rond, since the ectors hae the same magnitde and opposite directions, the rope was plled in opposite directions with eal force, and the match was a tie. In Rond, Team won since the magnitde of their force ector is greater than the magnitde of the force ector for Team. OU O ITIO 0 0 R D QU VTOR In ercises 5 8, se the diagram D shown at the right. H 5. hich ectors are parallel? Æ Æ Æ G, D, and 6. hich ectors hae the same direction? Æ Æ and D 7. hich ectors are eal? Æ Æ J and D 8. ame two ectors that hae the same magnitde bt different directions. GH Æ and J Æ TUG-O-R G In ercises 9 and 30, se the information below. The forces applied in a game of tg-of-war can be represented b ectors. The magnitde of the ector represents the amont of force with which the rope is plled. The direction of the ector represents the direction of the pll. The diagrams below show the forces applied in two different ronds of tg-of-war. Rond Rond Team Team Team Team I TUG-O-R In the game tg-of-war, two teams pll on opposite ends of a rope. The team that scceeds in plling a member of the other team across a center line wins. R center line center line 9. In Rond, are Æ and Æ parallel ectors? re the eal ectors? 30. In which rond was the otcome a tie? How do o know? Describe the otcome in the other rond. plain or reasoning. ee margin. es; no 9.7 Vectors

3 OO RROR XRI 3 34 tdents ma make mistakes adding ectors if the do not pa attention to the location of the initial point and the terminal point. Remind them to pa attention to the direction of the ector when representing it with components. ITIO OT XRI 4 45 The forces acting on a skdier are grait and air resistance. The amont of air resistance depends on the speed of the skdier and the cross-sectional area of the skdier. That is the reason some skdiers position themseles with their bodies parallel to arth. It increases their cross-sectional area, which in trn increases the air resistance. This slows their descent DO s. 3 34, check stdents drawings. 3. = 4, ; =, 4 ; + = 6, 5 3. = º6, ; = 5, 3 ; + = º, =, º4 ; = 3, 6 ; + = 5, 34. =, 3 ; =, º6 ; + = 3, º3 43. abot 6 mi/h; the speed at which the skdier is falling, taking into accont the breeze I YDIVIG skdier who has not et opened his or her parachte is in free fall. Dring free fall, the skdier accelerates at first. ir resistance eentall stops this acceleration, and the skdier falls at terminal elocit. R OU O ITIO ROGR RU op the ectors and. rite the component form of each ector. Then find the sm + and draw the ector DDIG VTOR et = 7, 3, =, 4, w = 3, 7, and z = º3, º7. ind the gien sm w 4, , w 0, z º, º z 4, º4 40. w + z YDIVIG In ercises 4 45, se the information and diagram below. skdier is falling at a constant downward elocit of 0 miles per hor. In the diagram, ector represents the skdier s elocit. stead breeze pshes the skdier to the east at 40 miles per hor. Vector represents the wind elocit. The scales on the aes of the graph are in miles per hor. 4. rite the ectors and in component form. = 0, º0 ; 4. et s = + = 40, 0. op the diagram and draw ector s. ee margin. 43. ind the magnitde of s. hat information does the magnitde gie o abot the skdier s fall? ee margin. 44. If there were no wind, the skdier wold fall in a path that was straight down. t what angle to the grond is the path of the skdier when the skdier is affected b the 40 mile per hor wind from the west? abot ppose the skdier was blown to 0 the west at 30 miles per hor. ketch a new diagram and find the skdier s DO new elocit. ee margin for graph. The new elocit is s = º30, º0. 0 0, hapter 9 Right Triangles and Trigonometr DO 578

4 Test reparation 47. hen k > 0, the magnitde of is k times the magnitde of and the directions are the same. hen k < 0, the magnitde of is k times the magnitde of and the direction of is opposite the direction of. Jstifications ma ar. 48. b. abot 0. mi/h at a direction of abot.3 north of east hallenge 46. riting rite the component form of a ector with the same magnitde as Æ J =, 3 bt a different direction. plain how o fond the ector. ample answer: 47. OGI ROIG et ector º3, ; methods ma ar. = r, s. ppose the horizontal and the ertical components of are mltiplied b a constant k. The reslting ector is = kr, ks. How are the magnitdes and the directions of and related when k is positie? when k is negatie? Jstif or answers. ee margin. 48. UTI-T RO motorboat heads de east across a rier at a speed of 0 miles per hor. Vector = 0, 0 represents the elocit of the motorboat. The crrent of the rier is flowing de north at a speed of miles per hor. Vector = 0, represents the elocit of the crrent. a. et s = +. Draw the ectors,, and s in a coordinate plane. ee margin for graph. b. ind the speed and the direction of the motorboat as it is affected b the crrent. c. ppose the speed of the motorboat is greater than 0 miles per hor, and the speed of the crrent is less than miles per hor. Describe one possible set of ectors and that cold represent the elocit of the motorboat and the elocit of the crrent. rite and sole a word problem that can be soled b finding the sm of the two ectors. R R In ercises 49 5, se the information below. 48. c. motorboat heads de s shown in the diagram below, a bmper car moes from point to point to east across a rier at a point and back to point. The car follows the path shown b the ectors. The speed of mi/h. The magnitde of each ector represents the distance traeled b the car from the crrent of the rier is initial point to the terminal point. flowing de north at a speed of.75 mi/h. ind the speed and the elocit of the motorboat as it is affected b the crrent of the rier. nswer: The speed is abot. mi/h, and the elocit is gien 60 ft b the ector + =, The answer to. 50 is a ector which gies the final position of the bmper car, while the answer to. 5 is a nmber which gies the total distance traeled b the bmper car. XTR HG 54 ft 0 mi/h crrent mi/h ample answer: =, 0 and = 0,.75 ; ee margin for word problem. 49. ind the sm of Æ and Æ. rite the sm ector in component form. 50. dd ector Æ to the sm ector from ercise 49. 0, 0 5. ind the total distance traeled b the car. abot 73 ft 5. ompare or answers to ercises 50 and 5. h are the different? ee margin. 8, Vectors 579 GIH RR XRI 47 The strctre of the langage in ercise 47 ma case difficlties for some nglish learners. Yo ma want to read the three informational sentences alod, offering eplanations as necessar, and then hae stdents carr ot the reasoning task independentl. 48a. s DDITIO RTI D RTHIG or esson 9.7: ractice eels,, and (hapter 9 Resorce ook, p. 03) Reteaching with ractice (hapter 9 Resorce ook, p. 06) ee esson 9.7 of the ersonal tdent Ttor or more ied Reiew: earch the Test and ractice Generator for ke words or specific lessons. 579

5 4 DIY HOOR QUIZ Transparenc ailable Draw ector Q in a coordinate plane. rite the component form of the ector and find its magnitde. Rond or answer to the nearest tenth. heck stdents' drawings.. (, 3), Q(, 4) 3, 7 ; 7.6. ( 4, 5), Q(3, ) 7, 6 ; 9. et a = 4,, b = 3,, and c = 7,. ind the gien sm. 3. b + c 4, 3 4. a + c, 0 IXD RVI 53. ince D and are rt. and all rt. are, D. ince is eilateral, Æ Æ. Æ Æ D, so D and b the lternate Interior ngles Thm. n eilateral triangle is also eianglar, so m = m = 60. the def. of and the sbstittion prop. of ealit, D. D b the ongrence Thm. orresponding parts of are, so D Æ Æ. the def. of midpoint, is the Æ midpoint of D. 53. ROO Use the information and the diagram to write a proof. (Reiew 4.5) GIV D and are right angles; Æ Æ is eilateral; D Æ ROV is the midpoint of D. UIG GR ind the ales of and. (Reiew 4.6) = 45, = 90 = 0, = 30 = 30, = 60 UIG GR ind the prodct. (kills Reiew, p. 798, for 0.) 57. ( + ) 58. ( + 7) 59. ( + ) 60. (7 + ) D QUIZ 3 elf-test for essons 9.6 and 9.7 XTR HG OT hallenge problems for esson 9.7 are aailable in blackline format in the hapter 9 Resorce ook, p. 0 and at DDITIO TT RRTIO. RITIG mmarize how to identif eal ectors and parallel ectors. ample answer: Two ectors are eal if the hae the same magnitde and direction. Two ectors are parallel if the hae the same or opposite directions.. T. a = 4.7, b = 9.4 m = 65. =, z = 7.0 m Y = m = 3.4, = 0.9 m = p = 7.7, =. m Q = 5 5. f = 4.7, m = 37.9 m G = l =.0, m = 4.0 m = 76.0 ole the right triangle. Rond decimals to the nearest tenth. (esson 9.6).. 3. Y b 46 5 a J 8 6 f 3 R p X G Draw ector Q Æ in a coordinate plane. rite the component form of the ector and find its magnitde. Rond or answer to the nearest tenth. (esson 9.7) 7. (3, 4), Q(º, 3) º5, º ; (º, ), Q(4, º3) 6, º5 ; (0, º), Q(3, 4) 3, 5 ; (, 6), Q(º5, º5) º7, º ; 3.0 Æ Æ. Vector T = 3, 8. Draw T in a coordinate plane and find its direction relatie to east. (esson 9.7) ee margin. abot 69 north of east et = 0, º5, = 4, 7, w = º, º3, and z =, 6. ind the gien sm. (esson 9.7). + 4, 3. + w, w º, º z, 6. + z 6, 3 7. w + z 0, 3 z Z m 580 hapter 9 Right Triangles and Trigonometr 580

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} = Explain your answer. UI RTI Vocabular heck oncept heck kill heck 2. n two of the points; as shown in ample 3 on p. 81, an two points on a line can be use to calculate the slope of the line. RTI ITIO 1. If ~ X, =, an X =, what