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1 The Egyptians used an ingenious pulley system to move the blocks of stone used in the construction of pyramids. To minimize the work needed to displace the blocks, they applied a force oriented at 26. (Work (Nm) is the scalar product of the force vector and the displacement vector.) 1500 N 26 200 m Rounded to the nearest Nm, what work is needed to displace a block of stone horizontally for a distance of 200 m, if the force applied to it is 1500 N oriented at 26o? 2 A) 131 511 Nm C) 228 768 Nm B) 194 076 Nm D) 269 638 Nm Vector u = (2, -5) makes an angle of 40 with vector v whose magnitude is 7.8 units. What is the scalar product (dot product) of u and v ? A) 27.0 units C) 32.2 units B) 27.4 units D) 42.0 units Q 3 R Given vector parallelogram PQRS. Which of the following statements is FALSE? P 4 A) PQ QR PR B) RP SP RS C) PS SR RP D) SQ QR RS O S Giv en the three vectors u, v, and w. y v = (-2, -3) w u and w are represented in the Cartesian u plane at right : Which of the following statements is x TRUE? 5 A) v and - u are opposite. B) u and v are equivalent. C) w and ( v + w ) are orthogonal. D) u and 3 v are collinear. u Given vectors u and v shown here. v Which of the following vectors represents the resultant, r, of u v ? 6 A) r C) r B) r D) r The magnitudes of two vectors are 12 and 16 respectively, and their directions differ by 60 degrees. What is the magnitude of the resultant of these two vectors? 7 An airplane flying East at 150 km/h encounters a 50 km/h wind blowing in a 30 East of North direction. What will be the airplane's resultant velocity? 8 Given u (3, 2), and v (1, - 4) What are the components of the resultant of the following vector operation? u2 v 9 Two unit vectors, u and v, form a 60 angle as shown. What is the magnitude of v the vector w if w u 3v ? 60 u 10 Consider the two vectors u and v. • The magnitude of u is 10 cm at an angle of 140. • The magnitude of v is 15 cm at an angle of 240. • c 2u 3v What is the magnitude of c ? 11 d The scalar product of vectors d and f is 138. Their respective magnitudes are 7 and 25 units. f What is the measure of angle between vectors d and f? 12 Given: u - 1, 1 and v 1, 2 . What are the components of u 2u 3v ? 13 Given u • v = 54. The magnitude of u is 12 cm and its direction is 70. The direction of v is 10. What are the components of v ? y u v x 14 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 hour. Unfortunately, there is a wind. If the pilot does not adjust his flight path, he will be at point (120, 160) after an hour. What is the speed of the wind? 15 What is the resultant vector of CA BA DB DA ? 16 In the polygon below, ABCG is a square. B C D and G are the midpoints of sides CG and AF, respectively. Side AB is parallel to side EF. D E Using the properties of vectors, show that CB AC FE GF GE . 17 A An airplane leaves airport A and must 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. F G N E O y S B (400, 200) During the flight, the plane encounters a steady wind. This wind is represented by the A (150, 125) x vector v = (20, -15). The pilot steers the plane so as to negate the effect of the wind. At what angle relative to the east should the pilot point the plane in order to reach airport B? 18 The vertices of quadrilateral ABCD on a Cartesian plane have the following coordinates: A(-6, 8), B(-9, 5), C(-3, 4), D(-3, 6) The following composite transformation is used to produce quadrilateral A"B"C"D": r0, 90 s x Using matrices, construct the image of quadrilateral ABCD by applying the given composite transformation. 19 Find the following matrix products. a) 20 4 0 1 -1 0 2 1 2 7 1 -2 4 7 3 2 4 3 -3 4 0 1 b) 3 -2 6 0 -1 5 0 4 1 4 2 2 6 0 5 3 1 1 Using matrices, construct the image of triangle ABC under the following composite transformation: rO, -90 hO, 2 y B A C 1 O 1 x