Correction key 1 D Example of an appropriate method /4 Let x, y be the wind vector (km) y 100, 150 x, y 10, 160 100 x, 150 y 10, 160 100 withot wind with wind 100 + x = 10 and x = 0 150 + y = 160 and y = 10 City A 100 x (km) Therefore x, y 0, 10 The speed of the wind: 0, 10 0 10. 36 Answer The wind speed is approximately.36 km/h.
3 Example of an appropriate method Draw the vector. Since the adjacent angles in a parallelogram are spplementary, w 10 180 60 = 10 Therefore, sing the Cosine Law v 60 w 3v w 3 1 3 1 cos 10 w 13 w 13 3.6 Answer The magnitde of the vector is 3.6 nits.
4 Example of an appropriate method Measre of angle A m A = 180 80 = 100 since two consective angle in a parallelogram are spplementary. A 100 F res 100 N 50 N 10 40 Resltant force (strength) F res 100cos 50 100 50 100 F res 119. 3 N Direction of resltant force sin 100 119.3 sin 50 sin θ 0.41 74 θ 4.38 The direction is 4.38 + 40, so abot 64.38.
Answer Tim mst apply a force of 119.3 N with a direction of 64.38. 5 A 6 D 7 D
8 Example of an acceptable soltion Given the resltant vector Magnitde of the resltant vector θ = 8 + 50 8 50 cos 45 1998.31 d 44.7 Direction v 44.7 sin 45 8 sin θ sin = 0.166 = 7.3 70-7.3 = 6.7 Answer: The reslting speed of the hot air balloon is 44.7 km/h in a direction of 6.7.
9 Example of an appropriate method Components of vector AB AB = (400 150, 00 15) = (50, 75) Components of the nknown vector v? AB B? AB v? = (50, 75) (0, -15) A v AB?? = (50 0, 75 + 15)? = (30, 90) Direction of the nknown vector 90 tan = 30 1.37? 90 km 30 km Answer: To the nearest degree, the pilot shold point the plane at an angle of 1 relative to the east in order to reach airport B. Note: Do not penalize stdents who did not rond off their final answer or who made a mistake in ronding if off.
Stdents who sed an appropriate method in order to determine the components of the nknown vector have shown that they have a partial nderstanding of the problem. 10 B 11 c = 45.96 cm Accept any answer between 45.9 cm and 46 cm. 1 C 13 Ronded to the nearest tenth v is 16.6. 14 A 15 B
16 Example of an acceptable soltion N W B a = 50 E C c = 00 b =? A AB c 00 km/h, North BC a 50 km/h, from Northwest AC b? In ABC b = a + c ac cos (B) b = 50 + 00 50 00 cos (45) b = 8 357. 8644 = 168.398 km Also
A sin B sin C sin a b c sin A sin 45 50 168.398 50 sin 45-1 sin A A sin 0.100 168.398 A 1.1195 A 1.1195 as shown or N 1.1195 E Answer: The reslting speed of the airplane is 168.4 km/h in a direction of 1.1 or N 1.1 E.
17 Example of an appropriate soltion Angle between vectors θ 70 10 60 v v cos θ 54 1 v 54 1 v 54 6 v cos 60 1 v 9 cm 10 9 cm x y x cos 10 9 x 9 cos 10 x 8.86 y sin 10 9 y 9 sin 10 y 1.56 Answer: The horizontal, x, component is 8.9. The vertical, y, component is 1.6. Do not penalize stdents who did not rond, or ronded incorrectly.
18 To the nearest nit, the magnitde is 8 nits. To the nearest degree, the direction is W76N or eqivalent.
Vectors 1 Qadrilateral RSTU is a parallelogram and M is the point of intersection of its diagonals. Antoine lists the following vector operation statements: R S M U T 1) ) 3) 4) 5) ST SR MU UT UR SM RS RU RT MT MR MS MU 0 SR ST RT Which of these statements are tre A) 1, and 3 only C), 4 and 5 only B) 1, and 5 only D) 1, 3 and 4 only A plane goes from city A to city B. In a Cartesian plane, city A is at the origin and city B has coordinates (100, 150). If there is no wind, the flight lasts one hor. Unfortnately, there is a wind. If the pilot does not adjst his flight path, he will be at point (10, 160) after an hor. What is the speed of the wind? 3 Two nit vectors, and v, form a 60 angle as shown. v What is the magnitde of the vector w if w 3v? 60
4 Peter and Marie are plling on an object. The forces they applied are 100 N and 50 N respectively bt in different directions: 40 and 10. The sitation is represented below. 100 N 50 N 10 40 Tim is going to replace them. What force mst Tim apply to prodce the same effect on the object (strength and direction)? 5 Given the following prism having a rectanglar base. E F A B H G D C Which vector is eqivalent to the resltant of the expression AD + HE + AE? A) DH C) FB B) BE D) BC
6 The Egyptians sed an ingenios plley system to move the blocks of stone sed in the constrction of pyramids. To minimize the work needed to displace the blocks, they applied a force oriented at 6. (Work (Nm) is the scalar prodct of the force vector and the displacement vector.) 1500 N 6 00 m Ronded to the nearest Nm, what work is needed to displace a block of stone horizontally for a distance of 00 m, if the force applied to it is 1500 N oriented at 6 o? A) 131 511 Nm C) 8 768 Nm B) 194 076 Nm D) 69 638 Nm 7 Given the three vectors, v, and w. v = (-, -3) and w are represented in the Cartesian plane below: y w x Which of the following statements is TRUE? A) v and - are opposite. B) and v are eqivalent. C) w and ( v + w ) are perpendiclar. D) and 3 v are collinear.
8 A hot air balloon is flying de soth at 50 km/h. N Sddenly, the wind starts blowing from the sotheast at 8 km/h. N-W N-E What is the reslting speed and direction of the hot air balloon? W S-W S-E E S 9 An airplane leaves airport A and mst fly to airport B. In the Cartesian plane on the right, these airports are represented by points A and B respectively. The scale of the graph is in kilometres. y O N S E Dring the flight, the plane enconters a steady wind. This wind is represented by the vector v = (0, -15). A (150, 15) B (400, 00) x The pilot steers the plane so as to negate the effect of the wind. To the nearest degree, at what angle relative to the east shold the pilot point the plane in order to reach airport B? 10 Given vectors = (-3, 9), v = (6, ), w = (6, -18) and k 0. Which of the following statements is FALSE? A) k kv k v B) k v k w = k v w C) and w are collinear. D) and v are orthogonal.
11 Consider the two vectors and v. The magnitde of is 10 cm at an angle of 140. The magnitde of v is 15 cm at an angle of 40. c 3v What is the magnitde of c? 1 The magnitdes of two vectors are 1 and 16 respectively, and their directions differ by 60 degrees. What is the magnitde of the resltant of these two vectors? A) -96 C) 4.34 B) 14.4 D) 96 13 Given vectors and v where: = AB with A(-5, 7) and B(3, -5) v = (6, 3) Find + v. Rond yor answer to the nearest tenth.
14 An airplane flying East at 150 km/h enconters a 50 km/h wind blowing in a 30 East of North direction. What will be the airplane's resltant velocity? A) 180 km/h [E 14 N] C) 00 km/h [N 30 E] B) 195 km/h [E 7 N] D) 13 km/h [E 19 S] 15 Given vectors and v shown below. v Which of the following vectors represents the resltant, r, of v? A) r C) r B) r D) r 16 An airplane is flying de north at 00 km/h. N Sddenly, the wind starts blowing from the northwest at 50 km/h. W N-W N-E E What is the reslting speed and direction of the airplane? S-W S S-E
17 Given v = 54. The magnitde of is 1 cm and its direction is 70. The direction of v is 10. y v x What are the components of v, to the nearest tenth? 18 On a compter screen, an alien ship was travelling at a very rapid speed. When it reached point A(3, -), it sddenly exploded with one piece moving to point B(-1, 3) and the other to point C(5, 1). N B(-1, 3) W E v C(5, 1) S A(3, -) What is the sm of vectors and v? Give the magnitde of the resltant vector to the nearest nit, and its direction to the nearest degree.