Scale Drawings 1. Measure each side and angle, and sketch the polygon using the scale 1 cm represents 2.5 cm.

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1 Chapte 6 Peequisite Skills BLM Scale Dawings 1. Measue each side and angle, and sketch the polygon using the scale 1 cm epesents 2.5 cm. 6. Use the cosine law to find the length of side s. 7. Use the cosine law to find the measue of B. 2. Decide on an appopiate scale fo dawing a scale diagam of each line segment on an eighth of a sheet of lette pape. Daw each scale diagam. a) 500 km 750 mm c) 23 cm d) 7000 km Tansfomations of Angles 3. The initial am of each angle is on the positive x-axis. Daw each angle. Then, find the measue of the smalle angle between the positive y-axis and the teminal am. a) an angle in standad position measuing 120 teminal am 17 below negative x-axis c) teminal am is in the fouth quadant and 60 fom the negative y-axis. Sine and Cosine Laws 4. Use the sine law to find the length of side e. 5. Use the sine law to find the measue of Q. 8. a) Given ΔPQR with P = 25, Q = 30, and p = 52 m, use the sine law to calculate the length of side q. Given ΔXYZ with x = 6.5 cm, y = 7.3 cm, and Y = 84.3, use the sine law to calculate the measue of Z. c) Given ΔABC with a = 21 cm, b = 23 cm, and c = 18 cm, use the cosine law to calculate the measue of A. 9. Tenzin measues the angle of elevation to an aiplane and finds it to be 54. Tenzin is 3.5 km fom Patheep, who measues the angle of elevation to the aiplane in the same diection and finds it to be 38. a) How fa is each peson fom the aiplane? What is the height of the aiplane? Numbe Popeties 10. Simplify each expession using the commutative, associative, and distibutive popeties whee a, b, c R. a) a (a b (2 1) c (2 b 2 c) a 4 ( 1)(a 4) d) a(b c)(b c) Calculus and Vectos 12: Teache's Resouce BLM 6-1 Chapte 6 Peequisite Skills Copyight 2008 McGaw-Hill Ryeson Limited

2 6.1 Intoduction to Vectos BLM Explain whethe each quantity is a vecto o a scala. a) tempeatue volume c) momentum d) weight e) mass f) fiction 6. Name all the equivalent vectos in each diagam. a) 2. Descibe the magnitude and diection of each vecto. Descibe the vecto using wods and symbols. a) 3. Convet each tue beaing to its equivalent quadant beaing. a) c) 235 d) 270 e) 310 f) Convet each quadant beaing to its equivalent tue beaing. a) N40 E S65 E c) S32 W d) N27 W e) S54 W f) N11 W 5. In you notebook, edaw the vecto shown. Then, daw uu a) a vecto paallel to AB uu a vecto opposite to AB uu c) a vecto equivalent to AB 7. State the opposite of each vecto. a) 300 km noth 700 N on a beaing of 120 c) 180 km/h on a quadant beaing of S25 W 8. Use an appopiate scale to daw each vecto. Label magnitude, diection, and scale. a) displacement of 50 km west velocity of 8 m/s on a beaing of 140 c) foce of 1200 N downwad d) acceleation of 240 m/s 2 on a quadant beaing of N40 W e) velocity of 7 km/h at 25 to the hoizontal 9. ABCD is a paallelogam with diagonals AC and BD that intesect at E. Assuming the popeties of a paallelogam, state, whee possible, an equivalent vecto to each of the following. uu a) AB uu c) DB uu AC uu d) EB Calculus and Vectos 12: Teache's Resouce BLM 6-2 Section 6.1 Intoduction to Vectos Copyight 2008 McGaw-Hill Ryeson Limited

3 6.2 Addition and Subtaction of Vectos BLM Daw a diagam to illustate each vecto sum o diffeence. a) 4. Given the vectos a, b, and c, daw a diagam of each expession. c) d) a) a a c) a b c b c b c d) a b c 5. The diagam shows thee conguent equilateal tiangles. 2. Expess the shotest vecto in each diagam as the sum o diffeence of the othe two vectos. a) 3. Given the vectos a, b a b and ( a b ) c., and c, constuct Expess uu each uu diffeence as a single vecto. a) AB uu AE uu uu uu AB uu AE ED c) BD BE uu uu d) CD BD 6. A hang glide is tavelling hoizontally with an acceleation of 15 m/s 2. The foce of gavity is acting on the hang glide at 9.8 m/s 2 vetically downwad. a) Daw a scale diagam showing the acceleation vectos and the esultant acceleation of the hang glide. Detemine the magnitude and diection of the esultant acceleation of the hang glide. 7. An aiplane leaves the aipot tavelling N30 W at 720 km/h. Afte 1 h, the aiplane then tuns noth and tavels anothe 1.5 h at 850 km/h. What is the displacement of the aiplane afte 2.5 h? Calculus and Vectos 12: Teache's Resouce BLM 6-3 Section 6.2 Addition and Subtaction of Vectos Copyight 2008 McGaw-Hill Ryeson Limited

4 6.3 Multiplying a Vecto by a Scala BLM Copy the vecto v shown onto gid pape. 4. Two vectos u and v make an angle of 40 with each othe. Constuct each vecto sum o diffeence. On the same gid, daw epesentatives of the following vectos. a) 3 v 0.5 v c) 2 v d) v 2. Simplify each of the following algebaically. a) a 2 a 4 a 7 u 5 v 2 u 8 v c) 2( u v ) 3( u 2 v ) d) 7 u 5 v 2( u v ) 2 u e) 3( u v ) 2( u v ) f) 6( u 2 v ) 5( u 3 v ) 3. Fo the vectos a a) 2 a 3 b c) a 2 b d) a 3 b and b shown, daw a) u 2 u c) 2 u 2 v 2 v v 5. In paallelogam ABCD, uu opposite uu sides ae paallel and equal, BP = PA, and uuu uu uu uuu AQ = QD. Let BP = u and AQ = v. Expess the following vectos in tems of u and uuu v. a) AD uu PA uu c) CD uu d) PQ uu e) u BD u f) PD uu g) AC Calculus and Vectos 12: Teache's Resouce BLM 6-5 Section 6.3 Multiplying a Vecto by a Scala Copyight 2008 McGaw-Hill Ryeson Limited

5 6.4 Applications of Vecto Addition BLM Detemine the esultant of each vecto sum. a) 54 km/h west and then 23 km/h south 120 m/s west and then 75 m/s east c) 64 m hoizontally and then 23 m vetically d) 12 N at 30 and then 7 N at 170 e) 90 km N45 E and then 54 km S30 E 6. A community cente plans to install a new basketball hoop on the side of the building. The hoop and backboad have a combined weight of 196 N and ae suppoted by a bace and a wie. Detemine the tensions in the wie and the bace. 2. Find the magnitude of the esultant of the fou foces. 3. Find the magnitude of the esultant vecto of each of the following systems of foces. a) 22 N and 26 N acting at an angle of 47 to one anothe 117 N and 223 N acting at an angle of 140 to one anothe 4. A 120 N sign is hanging fom two chains attached to a ceiling as shown. a) Daw the vecto diagam that illustates this situation. Detemine the tensions in the chains. 5. A sign weighing 98 N is suspended fom the middle of a 4 m long chain. The ends of the chain ae attached to a ceiling at points 3 m apat. Detemine the tensions in the chains. 7. An aiplane is flying at 560 km/h on a heading of 340. The wind is blowing at 140 km/h fom the east. a) Daw a vecto diagam of this situation. Detemine the gound velocity of the aiplane. 8. A sailo wishes to sail to a pot that is located S25 E of his pesent position. The aveage speed of his sailboat is 23 knots and the wind is blowing fom N85 E at 12 knots. a) Daw a vecto diagam of this situation. In which diection should the sailo sail? c) What will the sailboat s goundspeed be? 9. A canoeist leaves a dock and paddles he canoe at an angle acoss a ive. The cuent is flowing at 3 km/h. The esulting velocity of the boat is 5.4 km/h downsteam, in a diection that foms a 15 angle with the adjacent shoe. a) Daw a vecto diagam of this situation. Detemine the canoeist s velocity elative to the wate. c) How fa downsteam will she be in 20 min? Calculus and Vectos 12: Teache's Resouce BLM 6-6 Section 6.4 Applications of Vecto Addition Copyight 2008 McGaw-Hill Ryeson Limited

6 6.5 Resolution of Vectos Into Rectangula BLM Components 1. Detemine the hoizontal and vetical components of each foce. a) magnitude of 570 N, θ = 37 counteclockwise fom the hoizontal magnitude of 29 N, θ = 52 clockwise fom the hoizontal c) magnitude of 1250 N, θ = 28 clockwise fom the vetical d) magnitude of 375 N, θ = 14 counteclockwise fom the vetical 2. A socce ball is kicked and tavels with an aispeed of 4 m/s at an angle of 15 fom the hoizontal. Detemine the ate of climb and the hoizontal goundspeed of the socce ball. 3. A jet takes off at a velocity of 500 km/h at an angle of 20 fom the hoizontal. a) Find the ate of climb of the jet. Find the hoizontal goundspeed of the jet. 4. A woman is pulling on a ope attached to a toboggan with a 370 N foce at an angle of 35 to the hoizontal. Find the magnitude of the foce pulling the sled fowad and the magnitude of the foce pulling the sled upwad. 5. A cuise ship is being pulled into a dock using two opes, as shown in the diagam. Find the magnitude of the foce in each ope if a esultant foce of N is needed to pull the cuise ship at the desied speed. 6. The wind exets a foce of 230 N pependicula to the sail of a boat. The sail is set 12 off the cente of the boat. Resolve this foce into two vecto components, one paallel and one pependicula to the keel of the boat. 7. A tuck weighing N is esting on a amp that is inclined at an angle of 15 to the hoizontal. Resolve the weight of the tuck into the ectangula components keeping it at est. 8. A ope attached to a box is being used to dag it up a amp. A 130 N foce is applied to the box at an angle of 35 to the amp. a) Find the magnitude of the foce in the diection of motion of the box. Round you answe to the neaest tenth of a newton. Find the magnitude of the foce pependicula to the diection of motion of the box. Round you answe to the neaest tenth of a newton. 9. A foce of 50 N is esolved into two vecto components with magnitudes of 24 N and 41 N. Find the angle each vecto component makes with the 50 N foce. 10. A jet s takeoff velocity has a hoizontal component of 500 m/s and a vetical component of 85 m/s. Detemine the esultant velocity of the jet. 11. An aiplane tavelling with an aispeed of 725 km/h is at an angle of 20 to the hoizontal. Find the hoizontal goundspeed and the ate of climb of the aicaft, both in kilometes pe hou. Calculus and Vectos 12: Teache's Resouce Copyight 2008 McGaw-Hill Ryeson Limited BLM 6-7 Section 6.5 Resolution of Vectos Into Rectangula Components

7 Chapte 6 Review BLM Intoduction to Vectos 1. Convet each tue beaing to its equivalent quadant beaing. a) c) 280 d) Use an appopiate scale to daw each vecto. Label the magnitude, diection, and scale. a) a foce of 120 N downwad a velocity of 85 km/h at a tue beaing of 085 c) a velocity of 8 N at a quadant beaing of N70 W 6.2 Addition and Subtaction of Vectos 3. The diagam uu shows uu a ectangula uuu pism. Let AE = a, AB = b, and AD = c. Name the esulting vecto that each sum o diffeence epesents. 6.3 Multiplying a Vecto by a Scala 5. Suppose i, j, and k ae vectos of magnitude 1. All vectos ae pependicula to each othe. Sketch each of the following linea combinations of i, j, and k. a) i 2 j 2 k 2 i 3 j k c) 3 i j k d) i 2 j k 6.4 Applications of Vecto Addition 6. A sled is being towed by two sled dogs. The tow opes make angles of 25 and 20 with the diection of motion of the sled. If the esultant foce has a magnitude of 950 N, find the magnitude of the foce that each ope exets on the sled. a) a c) a e) c b c a b d) c f) a c b b 7. A mass of 150 kg is suspended fom the uppe end of a pole that is inclined 35 to the vetical. The pole is suppoted by a cable that makes an angle of 60 with its uppe end. Find the tension in the cable and the compession along the pole. 4. Find the esultant of each pai of vectos. a) F 1 = 85 N and F 2 = 125 N acting at an angle of 60 to each othe. F 1 = 15 N and F 2 = 7 N acting at an angle of 140 to each othe. c) F 1 = 30 N and F 2 = 65 N acting at an angle of 30 to each othe. 6.5 Resolution of Vectos Into Rectangula Components 8. A box with a mass of 275 N ests on a fictionless amp inclined at an angle of 25 to the level gound. What foce must be applied to the box at an angle of 45 to the amp so that it emains at est? Calculus and Vectos 12: Teache's Resouce BLM 6-8 Chapte 6 Review Copyight 2008 McGaw-Hill Ryeson Limited

8 Pactice Test BLM Fo questions 1 2, select the best answe. 1. Which quadant beaing is equivalent to a 169 tue beaing? A N11 W B S11 E C N79 E D S11 W 2. Which tue beaing is equivalent to a S53 W quadant beaing? A 217 B 127 C 307 D Use an appopiate scale and daw each vecto. Label the magnitude, diection, and scale. a) acceleation of 35 m/s 2 at a quadant beaing of S17 W momentum of 47 N at a tue beaing of The diagam shows two squaes. Name a single vecto equivalent to each expession. uu uu a) FA uu FE uu uu EB uu EF c) AE ED uu u AF u CD 5. Equilateal tiangle DEF joins the midpoints of equilateal tiangle ABC. Wite an expession that is equivalent to each uu single vecto. uu a) AF in tems of AC uu uu uu AE in tems of AB and BC uu uu uu c) AB in tems of AF and EC 6. Two opposing hockey playes paticipate in a face-off. One playe hits the puck with a foce of 225 N paallel to the boads, and the second playe hits the puck with a foce of 315 N pependicula to the boads. Calculate the magnitude and diection of the esulting foce. 7. If the wind is blowing fom the east at 87 km/h and an aiplane is on a heading of 225 with an aispeed of 750 km/h, find the gound velocity of the aiplane and the diection of flight. 8. A box that weighs 15 N ests on a fictionless amp inclined at an angle of 15 to the hoizontal. Find the magnitude of the foce diected up the amp that must be applied on the box to keep it at est. 9. A cate that weighs 75 N ests on a fictionless amp inclined at an angle of 35 to the hoizontal. What foce must be applied at an angle of 15 to the amp so that the cate emains at est? 10. A pilot maps out he flight plan and detemines that to each he destination on time he plane must tavel S15 E at 720 km/h. If the wind is fom S35 W at 130 km/h, what heading and aispeed should the pilot set to each he destination on time? 11. A foce of magnitude 15 N is the esultant of two foces, one which has a magnitude of 11 N and acts at 25 to the esultant. Find the magnitude and diection of the othe foce. 12. Campes often hang thei food between two tees. A food bag that weighs 435 N and is hung fom opes between two tees that ae 6 m apat. One ope is 4 m long and one ope is 5 m long. The opes attach to the tees in the same hoizontal plane. Find the tension in each ope. Calculus and Vectos 12: Teache's Resouce BLM 6-10 Chapte 6 Pactice Test Copyight 2008 McGaw-Hill Ryeson Limited

9 Chapte 6 Pactice Mastes Answes BLM (page 1) Peequisite Skills 1. c) 180 km/h on a quadant beaing of N25 E 8. Diagams may vay. a) 2. Answes may vay. Fo example: a) 1 cm : 100 km 1 cm : 150 mm c) 1 cm : 5 cm d) 1 cm : 1000 km 3. a) c) cm cm a) 61.5 cm 33.3 c) a) Patheep: m; Tenzin: m m 10. a) 2b 0 c) 8 d) ab 2 2abc ac Intoduction to Vectos 1. a) scala scala c) vecto d) vecto e) scala f) vecto 2. a) 180 N to the ight 50 km/h at 30 above the hoizontal 3. a) N65 E S60 E c) S55 W d) W e) N50 W f) N17 E 4. a) c) 212 d) 333 e) 234 f) Diagams may vay. Fo example, in the u uu uu diagam shown, IJ is paallel to AB, KL uu uuu is opposite to AB, and MN is equivalent uu to AB. c) d) e) uu 9. a) DC uu none c) none d) DE 6.2 Addition and Subtaction of Vectos 1. a) c) d) uu 6. a) AB uu uuu DE = BA u u uu FB = EC uu = ED uu uu, BC uu uu = FE uu uu, CD u u, EF = CB, FA = CD, uu uu, BF = CE uu uu uuu uu uuu AB = DC, BA = CD, AD uuu uu uu uu uu uu DA = CB, DE = EB, BE = ED, uu uu uu uu AE = EC, CE = EA 7. a) 300 km south 700 N on a beaing of 300 uu = AF uu = BC,, 2. a) w = v u uu QR uu = PR uu PQ Calculus and Vectos 12: Teache's Resouce BLM 6-12 Chapte 6 Pactice Mastes Answes Copyight 2008 McGaw-Hill Ryeson Limited

10 Chapte 6 Pactice Mastes Answes BLM (page 2) Multiplying a Vecto by a Scala 1. Diagams may vay. Fo example, 4. a) 2. a) 7 a d) 7 u 3. a) 7 v 5 u e) u 13 v 5 v c) u f) u 8 v 27 v c) c) d) d) uu 5. a) EB 6. a) uu BD uu c) ED uu d) CB 17.9 m/s 2, 33.2 fom the hoizontal km N10.7 W Calculus and Vectos 12: Teache's Resouce BLM 6-12 Chapte 6 Pactice Mastes Answes Copyight 2008 McGaw-Hill Ryeson Limited

11 Chapte 6 Pactice Mastes Answes BLM (page 3) 4. a) 7. a) Diagams may vay. Fo example, km/h at a beaing of a) Diagams may vay. Fo example, c) S54.4 E c) 15.9 knots 9. a) Diagams may vay. Fo example, 5. a) 2 v e) 2 u 2 v u f) u c) 2 u 2 v d) u g) 2 v v 2 u 6.4 Applications of Vecto Addition 1. a) 58.7 km/h at S66.9 W 45 m/s west c) 68.0 m at 19.8 above the hoizontal d) 8.0 N at N64.1 E e) 92.2 km at N79.5 E N N53 W o a) 44 N 153 N 4. a) Diagams may vay. Fo example, u u T 1 = N, T 2 = 41.7 N u N u 6. T 1 = N, T 2 = 52.5 N 2.6 km/h at an angle of 32.2 with the adjacent shoe c) 1.7 km 6.5 Resolution of Vectos Into Rectangula Components u u 1. a) F = N; F = N h v u u F = 17.9 N; F = 22.9 N h v u u c) F = N; F = N h v u u d) F = 90.7 N; F = h v 2. ate of climb: 1.0 m/s; hoizontal goundspeed: 3.9 m/s 3. a) km/h km/h 4. fowad: u N; upwad: uu N 5. R 1 = N; R 2 = N 6. paallel: N; pependicula: 47.8 N 7. vetical: N; hoizontal: N 8. a) N 74.6 N N is 28.4 fom 50 N; 24 N is 54.5 fom 50 N Calculus and Vectos 12: Teache's Resouce BLM 6-12 Chapte 6 Pactice Mastes Answes Copyight 2008 McGaw-Hill Ryeson Limited

12 Chapte 6 Pactice Mastes Answes BLM (page 4) m/s at 9.6 fom the hoizontal 11. ate of climb: km/h; hoizontal goundspeed: km/h Chapte 6 Review 1. a) S21 E N64 E c) N80 W d) S22 W 2. Diagams may vay. a) c) d) c) N; N 7. tension: N; compession: N N Pactice Test 1. B 2. D 3. a) uu 3. a) AF uu uu AC uu uu c) DE uu d) BD e) ED f) BE 4. a) N 10.6 N c) 92.2 N 5. a) uu 4. a) EA u 5. a) 1 u 2 AC uuu BA uu AB uu c) AC 1 uu 2 BC uu c) 2AF uu 2EC N, 54.5 fom the diection of the boads km/h at a beaing of N N 10. S23.9 E, km/h N, 42.7 fom esultant N and N Calculus and Vectos 12: Teache's Resouce BLM 6-12 Chapte 6 Pactice Mastes Answes Copyight 2008 McGaw-Hill Ryeson Limited