Sides and Angles of Right Triangles 6. Find the indicated side length in each triangle. Round your answers to one decimal place.

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1 Chapte 7 Peequisite Skills BLM Convet a Beaing to an Angle in Standad Position 1. Convet each beaing to an angle in standad position on the Catesian gaph. a) c) 215 d) 295 e) N40 W f) S65 E g) N15 E h) S20 W Distance Between Two Points 2. Find the distance between the points in each pai. a) A(5, 4), B( 3, 2) C(1, 3), D( 3, 2) c) E( 1, 2), F( 6, 3) d) G(0.1, 0.2), H( 0.1, 0.2) Popeties of Addition and Multiplication 3. Veif each statement. a) = = c) ( ) + 30 = 20 + ( ) d) (2 5) 3 = 2 (5 3) e) 20(5 + 3) = 20(5) + 20(3) f) Solve Sstems of Equations 4. Solve each linea sstem b elimination. a) = = 12 3m + 4n = 1 4m 5n = 22 c) 5p + 3q = 19 2p 5q = 11 d) = = 0.8 Evaluating the Sine and Cosine of an Angle 5. Evaluate using a calculato. Round ou answes to one decimal place. a) sin 31 cos 17 c) cos 210 d) sin 285 e) sin 173 f) cos 245 Sides and Angles of Right Tiangles 6. Find the indicated side length in each tiangle. Round ou answes to one decimal place. a) 7. Epess h in tems of the given infomation. a) c) Simplifing Epessions 8. Epand and simplif. a) (2 7) 2 (4 5) (4 + 5) c) ( )( ) d) ( ) 3 Calculus and Vectos 12: Teache's Resouce BLM 7-1 Chapte 7 Peequisite Skills Copight 2008 McGaw-Hill Reson Limited

2 7.1 Catesian Vectos BLM Epess each vecto in tems of i and j. a) [ 2, 0] [0, 3] c) [3, 2] d) [ 1, 6] e) [2, 7] f) [0, 8] g) [1.4, 4.7] h) [ 3.2, 8.7] 2. Epess each vecto in the fom [a, b]. a) 3 i + 2 j 4 j c) 7 i + 3 j d) 9 i e) 2 i + 5 j f) 3.1 i 7.2 j g) 6.5 i j h) 8.9 i 3.2 j 3. Wite the coodinates of each Catesian vecto and detemine the magnitude. 5. If u = [ 3, 5] and v = [ 3, 5]. a) u + v 5 u v c) 3 u + 4 v d) 7 v + 6 i 8 j 3 u e) v f) 3 v 2 u 6. Wite each foce as a Catesian vecto. a) 750 N applied 45 to the hoiontal 215 N applied 68 to the vetical c) 450 N applied upwads d) 17 N applied downwads e) 1000 N east f) 80 N west 7. An aicaft is tavelling at 750 km pe hou at an angle of 35 to the level gound below. Find the foce in component fom as a Catesian vecto. 8. A mom is pulling a sled eeting a foce of 220 N along a ope that makes an angle of 20l to the hoiontal. Wite this foce in component fom as a Catesian vecto. A 5 B 9. Let a = ( 2, 5) and b = (5, 7). a) Plot the two vectos. Which is geate: a + b o a + b? 5 5 D C 5 4. Given the vecto v = [ 3, 7]. a) State the vetical and hoiontal vecto components of v. Find two unit vectos that ae collinea with v. Calculus and Vectos 12: Teache's Resouce BLM 7-2 Section 7.1 Catesian Vectos Copight 2008 McGaw-Hill Reson Limited

3 7.2 Dot Poduct BLM Calculate the dot poduct fo each pai. a) 4. Let a = [1, 2], b = [2, 5], and c = [4, 1]. Detemine each dot poduct. a) a ( b + c ) ( a + b ) c c) ( a + b ) ( a + c ) d) (3 a + 2 b ) (4 a b ) 5. Find the angle θ between each pai of vectos. a) u = [4, 6] and v = [ 3, 1] u = [3, 5] and v = [1, 8] 6. Fo u = [5, 7] and v = [ 4, 1], veif each equation. a) u v = v u v v = v 2 c) (2 u + 3 v )(2 u 3 v ) = 4 u 2 12 u v 9 v 2 2. Calculate the dot poduct fo each pai of vectos θ is the angle between the vectos when the ae placed tail to tail. a) u = 7, v = 12, and θ = 47 s = 520, t = 745, and θ = 135 c) q = 12, = 3, and θ = 3 d) m = 230, n = 170, and θ = Calculate the dot poduct of each pai of vectos. a) a = [5, 8], b = [ 2, 1] c = [ 1, 8], d = [3, 3] c) e = [2, 5], f = [ 3, 1] d) g = [7, 2], h = [ 3, 4] e) l = 2 i 3 j, m = 9 i + 4 j f) u = 6 i + 7 j, v = 3 i 2 j 7. Two vectos and 1 5 ae pependicula. Find the angle between and, if = Detemine the value of a so that u = [2, 3] and v = [a, 8] ae pependicula. 9. Detemine the value of k so that u = [5, k] and v = [9, 3] ae pependicula. 10. Detemine a value of t so that u = [9, t] and v = [ 16, t] ae pependicula. Calculus and Vectos 12: Teache's Resouce BLM 7-3 Section 7.2 Dot Poduct Copight 2008 McGaw-Hill Reson Limited

4 7.3 Applications of the Dot Poduct BLM Detemine the wok done b each foce F, in Newtons, fo each object moving along s. a) F = [3, 2], s = [1, 8] F = [8, 9], s = [ 3, 7] c) F = [ 2.1, 3.8], s = [9.2, 5.4] d) F = [17.2, 11.8], s = [21.8, 15.3] 2. Detemine the wok done b the foce F, in Newtons, fo each object moving along s. a) 4. Detemine the pojection of the fist vecto in the second. a) a = [6, 1], b = [3, 4] c = [6, 7], d = [3, 2] c) p = [ 4, 1], q = [2, 3] d) p = [6, 7], q = [3, 2] 5. DEF has vetices D( 3, 5), E(2, 3), and F(6, 7). Calculate DEF. 6. A cate on a amp is hauled 12 m up the amp unde a constant foce of 120 N applied at an angle of 45 to the amp. Find the wok done. 7. How much wok is done b the odel pushing a 85 kg peson up a 5 m amp inclined at an angle of 15 to the hoiontal? 3. Detemine the angle between the vectos in each pai. a) p = [6, 7] and q = [3, 2] = [ 1, 7] and s = [5, 4] c) e = [ 7, 8] and f = [9, 11] d) g = [ 4, 3] and h = [2, 6] 8. A stage lamp is dagged 15 m along level gound b a 120 N foce applied at an angle of 35 to the gound. It is then dagged up a 12m amp onto a stage b the same foce. If the amp is inclined at 15 to the gound. Find the total wok done. 9. A bo on a wagon pulled a distance of 35 m b a 27 N foce applied at an angle of 40 to the gound. The bo is then lifted a distance of 1.5 m and placed on a table b eeting a foce of 37 N. Find the total wok done. Calculus and Vectos 12: Teache's Resouce BLM 7-4 Section 7.3 Applications of the Dot Poduct Copight 2008 McGaw-Hill Reson Limited

5 7.4 Vectos in Thee Space BLM Daw the position vectos. a) [ 2, 3, 4] [2, 3, 1] c) [0, 2, 5] d) [4, 0, 5] e) [4, 2, 0] f) [ 1, 4, 3] 2. Epess each vecto as the sum of i, j, and k. a) [2, 1, 7] [ 4, 6, 5] c) [3, 8, 0] d) [2, 6, 1] 3. Epess each vecto in the fom [a, b, c]. a) 3 i 4 j + 5 k 2 i + 3 k c) 8 i + 9 j 4 k d) 8 j 7 k uuu 4. Daw vecto AB joining each pai of points. Then wite the vecto in the fom [a, b, c]. a) A(2, 1, 7) and B(0, 2, 1) A(0, 4, 2) and B( 3, 1, 0) c) A(5, 6, 3) and B(1, 3, 2) d) A(9, 7, 6) and B(4, 8, 9) 5. Evaluate each given the vectos a = [ 2, 1, 8], b = [3, 1, 2], and c = [2, 3, 4]. a) 3 b b c c) 2 a 3 c + 4 b d) ( a + b ) ( a + c ) e) b c f) a b c b 6. Let a = 3 i 2 j + 4 k, b = 7 i + 4 j k, and c = 2 i + 5 j + 9 k. a) ( a + b ) c ( a + b ) ( b + c ) c) (2 a 3 b ) (4 b + c ) d) 2 a (4 b 3 c ) 7. Detemine the values of k such that u and v ae othogonal. a) u = [2, k, 1] and v = [3, 2, 7] u = [2, 5, k] and v = [4, k, 3] c) u = [ 3, 1, k] and v = [4, k, k] 8. Find a vecto othogonal to each vecto. a) [2, 1, 7] [8, 3, 4] 9. Conside the vectos u = [3, 5, 8] and v = [3, 1, 2]. a) Find u v. Calculate the angle between u and v. 10. Find the angle θ between each pai of vectos. a) u = [3, 4, 7] and v = [2, 1, 1] u = [ 1, 5, 7] and v = [8, 3, 2] 11. Detemine the pojection of a and b. a) a = [2, 1, 3] and b = [1, 7, 6] a = [3, 4, 7] and b = [2, 1, 1] c) a = [9, 0, 8] and b = [5, 1, 3] d) a = [ 11, 15, 12] and b = [3, 2, 4] Calculus and Vectos 12: Teache's Resouce BLM 7-5 Section 7.4 Vectos in Thee Space Copight 2008 McGaw-Hill Reson Limited

6 7.5 Coss Poduct and Its Popeties BLM Detemine u! v. a) 4. Given a = [1, 2, 1], b = [2, 2, 1], and c = [2, 3, 4], evaluate each of the following: a) a! b! c a! b ( ) ( )! c c) a! c " a! b d) b! 3c e) ( a! c ) " b f) ( a! b ) " c g) a! b h) #a! ( b " c) $ % & 5. Detemine the angles between the vectos. a) a = [2, 1, 3], b = [5, 4, 3] a = [ 1, 4, 1], b = [3, 2, 4] 6. Detemine the aea of PQR with vetices of P(3, 2, 7), Q(2, 2, 3), and R(1, 1, 2). c) u = [2, 1, 7], v = [2, 1, 3] d) u = [ 3, 4, 7], v = [4, 3, 5] e) u = 3 i + 4 j! k, v = 5 i + j! 2 k f) u = 2 i! 3 j + 7 k, v =! i + j 2. Find a vecto pependicula to each of the following pais of vectos. Use the dot poduct to check ou answe. a) [5, 0, 1] and [ 2, 5, 8] [1, 4, 2] and [ 4, 9, 0] c) [ 1, 7, 2] and [9, 2, 5] d) [7, 10, 0] and [0, 0, 1] 3. Find a unit vecto pependicula to a = [6, 2, 3] and b = [5, 1, 4]. 7. Detemine the aea of ABC with vetices of A(1, 1, 1), B( 1, 2, 2), and C(3, 1, 3). 8. Detemine the aea of the paallelogam ABCD defined b the vetices A(2, 1, 5), B( 2, 7, 8), C(1, 3, 8), and D(4, 3, 7). 9. Detemine the aea of the paallelogam ABCD defined b the vetices A(2, 1, 1), B( 4, 2, 3), C(2, 3, 2), and D(8, 4, 2). 10. If u and v ae collinea, use components, to pove that u! v = Fo an vectos u and v, use components to pove that u! v = " v! u. Calculus and Vectos 12: Teache's Resouce BLM 7-7 Section 7.5 Coss Poduct and Its Popeties Copight 2008 McGaw-Hill Reson Limited

7 7.6 Application of the Dot Poduct and Coss Poduct BLM Given a = [2, 4, 5], b = [ 1, 3, 7], and c = [ 2, 7, 3], evaluate each epession. a) a b c a c b a c) b c d) c a b 2. Detemine the pojection, and its magnitude of u on v. a) u = [2, 1, 7], v = [ 7, 2, 6] u = [1, 7, 3]. v = [2, 3, 5] c) u = 7 i 6 j + 5 k, v = 3 i 2 j + k d) u = 2 i + 5 j + 7 k, v = i 4 j + 5 k 3. Detemine the wok done in the diection of tavel. u a) F u = [200, 150, 75], s = [2, 1.8] u F = [7, 9, 11], s = [ 3, 2. 1] c) F = 3 i + 9 j + 5 k, s = 2 i + 5 j + 3 k u d) F = 2 i 3 j + 7 k, s = 8 i + 2 k 4. Find the aea of the paallelogam with sides consisting of the vectos. a) a = [ 4, 5, 8], b = [1, 2, 3] a = [9, 5, 7], b = [3, 2, 5] 5. Find the aea of the tiangle with the given vetices. a) A(0, 2, 4), B(3, 2, 1), C(4, 2, 5) A( 2, 4, 5), B(1, 4, 2), C(7, 4, 9) 6. Detemine the volume of the paallelepiped detemined b the vectos. a) a = [2, 5, 8], b = [7, 2, 3], and c = [8, 2, 1] a = [1, 5, 9], b = [3, 4, 7], and c = [1, 0, 2] 7. Find the toque poduced b a cclist eeting a foce of 85 N on the pedal in the position shown in the diagam, if the shaft of the petal is 11 cm long. 8. A woman pushes he bab stolle a distance of 1500 m b a foce of 89 N applied at an angle of 35 to the oadwa. Calculate the wok done. 9. Detemine the wok done b gavit in causing a 45 kg child to slide down a 55 m slope, which has an angle of 47 to the hoiontal. 10. A 50 kg bo is located 12 m up a amp inclined at an angle of 25 to the hoiontal. Detemine the wok done b the foce of gavit as the bo slides to the bottom of the amp. Calculus and Vectos 12: Teache's Resouce BLM 7-8 Section 7.6 Applications of the Dot and Coss Poduct Copight 2008 McGaw-Hill Reson Limited

8 Chapte 7 Review BLM Conside the vecto v = [8, 7]. a) Wite v in tems of i and j. Wite vˆ. 2. Given u = [2, 1] and v = [5, 7], evaluate each epession. a) 2 v u v c) 3 u + 5 v d) 4 u 2 v 3. A boat is sailing at a speed of 34 km/h on a heading N15 E. A wind is blowing at 7 km/h fom the west. Detemine the gound velocit of the boat. Answe in Catesian fomat. 4. Calculate the dot poduct. a) c) u = [ 8, 9] and v = [1, 7] d) u = 5 i + 11 j and v = 7 i 2 j 5. If a = 12 and b = 12, and the dot poduct of the two vectos is 98, then find the angle θ between a and b. 6. Calculate the angle θ between the vectos. a) u = [ 1, 8] and v = [3, 5] u = [3, 7] and v = [6, 1] 7. Detemine the pojection of u on v. a) u = 95 and v = 275, and the angle between u and v is 13. u = [6, 5] and v = [7, 11] 8. Detemine the wok done b each F, in Newtons, fo an object moved along the vecto, d, in metes. a) F = [18, 23] and d = [9, 12] F = [500, 2700] and d = [150, 275] 9. Detemine uuu the eact magnitude of the vecto AB joining A(3, 8, 11) and B(7, 3, 15). 10. Given a = [2, 9, 7], b = [8, 7, 5], and c = [1, 3, 2], evaluate each epession. a) 7 a + 2 b + c a b c) a c 11. If u = [5, 7, 9] is othogonal to v = [1, t, 1], detemine t. 12. Detemine the aea of the paallelogam bound b the vectos a = [7, 11, 4] and b = [3, 2, 5]. 13. A 78 kg bo is located 2 m up a amp inclined at an angle of 23 to the hoiontal. Detemine the wok done b the foce of gavit as the bo stats to slide down the amp. 14. A 525 N foce is applied to the end of a 130 cm wench and makes an angle of 25 with the handle of the wench. What is the toque on the bolt at the othe end of the wench? Calculus and Vectos 12: Teache's Resouce BLM 7-9 Chapte 7 Review Copight 2008 McGaw-Hill Reson Limited

9 Chapte 7 Test BLM Evaluate, given vectos u = [2, 3] and v = [3, 1]. a) 3 u + 2 v u c) u v d) u + v 2. Given vectos a = i + 3 j 2 k and b = 2 i + j + 3 k, evaluate each of the following. a) a + b 3 a 2 b c) a d) a b e) 2 a b + 3 b a f) a b 3. Given c = [ 6, 4, 0], d = [0, 5, 7], and e = [3, 1, 2]. a) 10 c 10 d 2 c 3 d 2 c 4 e c) (3 d 4 e ) 2 c d) c d e) d e f) c c 4. The angle between two vectos u and v is 30. If u = 25, find the pojection of u on v. 5. Fo each of the following calculate the angle between the two vectos. a) [1, 2] and [ 3, 4] [ 1, 2] and [2, 1] c) [0, 1, 2] and [ 3, 1, 4] d) [1.5, 20, 0] and [ 20, 1, 10] 6. A tiangle has vetices A(2, 5, 1), B(4, 1, 3), and C( 5, 6, 4). Find the aea of the tiangle. 7. Calculate the aea of the paallelogam ABCD in each of the following. a) A(4, 6), B(5, 6), C(11, 15), and D( 2, 3) A(11, 15, 15), B( 2, 3,7), C(4, 6, 0), and D(5, 6, 8) 8. Find the volume of the paallelepiped defined b a = (1, 0, 4), b = (8, 7, 9), and c = (2, 3, 4). Calculus and Vectos 12: Teache's Resouce BLM 7-11 Chapte 7 Test 9. Calculate the wok done b a foce, F, acting though a displacement, d : a) F = [ 8, 5] and d = [7, 9] F = [7, 1, 4] and d = [ 3, 11, 7] 10. Calculate the wok done b a 10 N foce acting in the diection of (2, 3) in moving an object fom A( 3, 4) to B( 1, 6), assuming that distance is measued in metes. 11. A hoppe pushes a loaded supemaket cat of mass 20 kg up a 10 m amp inclined at 15 to the hoiontal. The shoppe then pushes the cat towads he ca, which is anothe 25 m awa. Calculate the total wok done in moving the cat. 12. Find the acute angle of a paallelogam if the sides ae 15 m and 9 m and the aea is 120 m A small plane A is fling on a beaing of 283 at 200 km/h and anothe, B is fling on a beaing 071 at 140 km/h. Calculate the diection and magnitude of the velocit of A elative to B. 14. A tanke is tavelling with a velocit of v = [30, 15] and a foklift on boad is taveling elative to the tanke with a velocit of v = 9 i + 7 j. A spide is cawling on the foklift with velocit v = i + 5 k elative to the foklift. Calculate the velocit and speed of the spide elative to the ocean. 15. A foce of 1550 N is applied to a scew b an hand wench with a handle length of 50 cm at an angle of 35 to the handle. Find the moment of foce about the scew. 16. If u = 1 and v = 2 and the angle between u and v is 60 evaluate (4 u + 5 v ) ( 2 u 3 v ). Copight 2008 McGaw-Hill Reson Limited

10 Chapte 7 Pactice Mastes Answes BLM (page 1) Peequisite Skills 1. a) c) 235 d) 65 e) 130 f) 335 g) 75 h) a) c) 50 d) a) [ 2, 4] [ 3, 2] c) [ 2, 3] d) [4, 3] 5. a) c) 0.87 d) 0.97 e) 0.12 f) a) a) c) a) c) d) Catesian Vectos 1. a) 2 i 3 j c) 3 i + 2 j d) i + 6 j e) 2 i 7 j f) 8 j g) 1.4 i j h) 3.2 i 8.7 j 2. a) [3, 2] [0, 4] c) [ 7, 3] d) [ 9, 0] e) [ 2, 5] f) [3.1, 7.2] g) [6.5, 4.7] h) [ 8.9, 3.2] 3. A) (3, 2); 13 B) ( 5, 2); 29 C) (1, 4); 17 D) ( 4, 3); 5 4. a) vetical: 7; hoiontal: 3 e.g., [ 6, 14], [3, 7] 5. a) [ 4,9] [ 14, 21] c) [5, 1] d) [8, 5] e) 17 f) a) [530.3, ] [ 199, 80.5] c) [0, 450] d) [0, 17] e) [1000, 0] f) [ 80, 0] 7. [614.36, ] 8. [206.7, 75.2] 9. a) 7.2 Dot Poduct 1. a) a) c) d) a) 2 27 c) 1 d) 29 e) 30 f) a) 2 9 c) 6 d) a) a = k = t = 12, Application of the Dot Poduct 1. a) c) 1.2 d) a) a) θ = 15.7 θ = c) θ = d) θ = a) [2.64, 3.52] [7.38, 4.92] c) [ 0.769, 1.154] d) [0.923, 0.615] J J J J 7.4 Vectos in Thee Space 1. a)! 4! 2 3 a ! b v a + b is geate Calculus and Vectos 12: Teache's Resouce BLM 7 13 Chapte 7 Pactice Mastes Answes Copight 2008 McGaw Hill Reson Limited

11 Chapte 7 Pactice Mastes Answes BLM (page 2) c) 4. a) [ 2, 3, 8] 5 A 7! 2!1 2 B d) [ 3, 3, 2] 4 e)! 5! 4 A!1 B! 2! 3! 2 4 c) [ 4, 9, 5] f) B! ! 3 6! 4!1 A! 3 2. a) 2 i j + 7 k 4 i 6 j + 5 c) 3 i + 8 j d) 2 i + 6 j k 3. a) [3, 4, 5] [2, 0, 3] c) [ 8, 9, 4] d) [0, 8, 7] k Calculus and Vectos 12: Teache's Resouce BLM 7 13 Chapte 7 Pactice Mastes Answes Copight 2008 McGaw Hill Reson Limited

12 Chapte 7 Pactice Mastes Answes BLM (page 3) d) [ 5, 1, 3] A B!8 5. a) [9, 3, 6] [1, 4, 6] c) [2, 15, 4] d) [1, 4, 6] e) 5 f) a) c) 573 d) a) k = 0.5 k = 4 c) k = 4, k = 3 8. a) [4, 8, 0] [1,0, 2] 9. a) a) a) [ 0.10, 0.73, 0.63] [3, 1.5, 1.5] c) [3, 0.6, 1.8] d) [ 1.6, 1.0, 2.1] 7.5 Coss Poduct and Its Popeties 1. a) ˆn 84.9 ˆn c) [ 10, 8, 4] d) [ 41, 13, 25] e) [ 7, 1, 17] f) [ 7, 7, 1] 2. a) [ 5, 42, 25] [18, 8, 25] c) [39, 23, 61] d) [ 1, 7, 0] 1 3. [11, 9, 16] a) [26, 21, 16] [22, 28, 10] c) [1, 3, 5] d) [ 33, 18, 30] e) 13 f) 13 g) 53 h) a) ! Application of Dot Poduct and Coss Poduct 1. a) c) 119 d) a) [ 7, 2, 6]; [2, 3, 5]; 34 38! 6! 9 c) [3, 2, 1]; d) [1, 4, 5]; a) 850 J 50 J c) 54 J d) 65 J 4. a) a) a) N m J J J Review 1. a) 8 i 7 1 j [8, 7] a) [10, 14] [ 3, 6] c) [31, 38] d) [ 2, 10] 3. [15.78, ] 4. a) c) 71 d) θ = a) θ = θ = a) nˆ 97 [7, 11] a) 438 J J a) [31,52, 41] 82 c) [3, 3, 3] 11. t = J N Test 1. a) [12, 7] 13 c) 3 d) a) [3, 4, 1] [ 1, 7, 12] c) 14 d) 1 e) 5 f) a) ( 60, 90, 70) 8 c) 8 d) ( 28, 42, 30) e) ( 3, 21, 15) f) nˆ 5. a) c) d) a) a) 101 J 60 J Calculus and Vectos 12: Teache's Resouce BLM 7 13 Chapte 7 Pactice Mastes Answes Copight 2008 McGaw Hill Reson Limited

13 Chapte 7 Pactice Mastes Answes BLM (page 4) J km/h west 14. (21,22,5); 30.8 km/h nˆ Calculus and Vectos 12: Teache's Resouce BLM 7 13 Chapte 7 Pactice Mastes Answes Copight 2008 McGaw Hill Reson Limited