Chapter 6 REVIEW. 6.1 Introduction to Vectors. 6.3 Multiplying a Vector by a Scalar. 6.2 Addition and Subtraction of Vectors

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1 Chapter 6 REVIEW 6.1 Introduction to Vectors 1. For which of the following situations would a vector be a suitable mathematical model? Provide a reason for your decision. a) A car is travelling at 70 km/h northeast. b) A boy is walking at 5 km/h. c) A rocket takes off at an initial speed of 800 km/h at 80 from the horizontal. d) An airplane is sighted 20 km away. e) A man s height is 180 cm. 2. Convert each true bearing to its equivalent quadrant bearing. a) 130 b) 080 c) Use an appropriate scale to draw each vector. Label the magnitude, direction, and scale. a) velocity of 140 km/h due west b) acceleration of 20 m/s 2 at a bearing of 105 c) force of 100 N upward 6.2 Addition and Subtraction of Vectors 4. The diagram shows a regular octagon. Write a single vector that is equivalent to each vector expression. a) HA AB b) GH FG c) FE BA d) GA EH DG H G A F B E C D 5. A camera is suspended by two wires over a football field to get shots of the action from above. At one point, the camera is closer to the left side of the field. The tension in the wire on the left is 1500 N, and the tension in the wire on the right is 800 N. The angle between the two wires is a) Draw a vector diagram of the forces, showing the resultant. b) Determine the approximate magnitude and direction of the resultant force. 6.3 Multiplying a Vector by a Scalar 6. Express each sentence in terms of scalar multiplication of a vector. a) An apple has a weight of 1 N, and a small car has a weight of N. b) A boat is travelling at 25 km/h northbound. It turns around and travels at 5 km/h southbound. c) Acceleration due to gravity on Earth is 9.8 m/s 2, and on the Moon it is 1.63 m/s ABCDE is a pentagon such that AB DC and AC 2 ED. Write each vector in terms of AB and AC. a) EC b) CE c) CB d) AE E A D B C 352 MHR Calculus and Vectors Chapter 6

6.4 Applications of Vector Addition 8. Find the resultant of each pair of vectors. a) b) c) 16.1 km 500 N 0 N 300 km/h 240 km/h N 295 300 N 25 12.7 km 9.

The other guy wire is inclined at 40 to the horizontal and is under 400 N of tension. Determine the magnitude and direction of the resultant force on the guy wires. 10. Three forces act on a body.

2 6.4 Applications of Vector Addition 8. Find the resultant of each pair of vectors. a) b) c) 16.1 km 500 N 0 N 300 km/h 240 km/h N N km 9. During a wind storm, two guy wires supporting a tree are under tension. One guy wire is inclined at an angle of 35 and is under 500 N of tension. The other guy wire is inclined at 40 to the horizontal and is under 400 N of tension. Determine the magnitude and direction of the resultant force on the guy wires. 10. Three forces act on a body. A force of 100 N acts toward the north, a force of 120 N acts toward the east, and a force of 90 N acts at N20 E. a) Describe a method for finding the resultant of these three forces. b) Use your method to determine the resultant. 6.5 Resolution of Vectors Into Rectangular Components 11. In basketball, hang time is the time a player remains in the air when making a jump shot. What component(s) does hang time depend on? Explain. 12. A 1000-kg load is suspended from the end of a horizontal boom. The boom is supported by a cable that makes an angle of 35 with the boom. a) What is the weight of the load? b) What is the tension in the cable? c) What is the horizontal force on the boom? d) What is the vertical equilibrant component of the tension in the cable? kg PROBLEM WRAP- UP CHAPTER A small plane is heading north at 180 km/h. Its altitude is 2700 m. a) Draw a labelled scale vector diagram of the effects of a 90-km/h wind from the west. Include the resultant in your diagram. b) Determine the ground velocity of the airplane. c) The airplane descends to 2000 m over a period of 2 min, still flying at the same groundspeed. What is the horizontal component of the change in displacement? d) The airplane then enters turbulent air, which is falling at 30 km/h. The turbulence does not affect the groundspeed of the airplane. What is the airplane s resultant velocity? e) The airplane then enters more turbulent air. The air mass is moving upward at 20 km/h and moving N30 W at 60 km/h, while the plane maintains its airspeed. Determine the vectors that represent the turbulent air and the resultant velocity of the airplane. Review MHR 353

3 Chapter 7 REVIEW 7.1 Cartesian Vectors 1. Consider the vector v [ 6, 3]. a) Write v in terms of i and j b) State two unit vectors that are collinear with v. c) An equivalent vector AB has initial point A(2, 9). Determine the coordinates of B. 2. Given u [5, 2] and v [8, 5], evaluate each of the following. a) 5u b) u v c) 4u 2v d) 3u 7v 3. An airplane is flying at an airspeed of 345 km/h on a heading of 040. A wind is blowing at 18 km/h from a bearing of 087. Determine the ground velocity of the airplane. Include a diagram in your solution. 7.2 Dot Product 4. Calculate the dot product of each pair of vectors. Round your answers to two decimal places. a) v Which vectors in question 5 are orthogonal? Explain. 7.3 Applications of the Dot Product 7. Two vectors have magnitudes of 5.2 and 7.3. The dot product of the vectors is 20. What is the angle between the vectors? Round your answer to the nearest degree. 8. Calculate the angle between the vectors in each pair. Illustrate geometrically. a) a [6, 5], b [7, 2] b) p [ 9, 4], q [7, 3] 9. Determine the projection of u on v. a) u 56, v 100, angle θ between u and v is 125 b) u [7, 1], v [9, 3] 10. Determine the work done by each force, F, in newtons, for an object moving along the vector d, in metres. a) F [16, 12], d [3, 9] b) F [200, 2000], d [3, 45] 11. An electronics store sells 40-GB digital music players for $229 and 80-GB players for $329. Last month, the store sold 125 of the 40-GB players and 70 of the 80-GB players. b) u a) Represent the total revenue from sales of the players using the dot product. b) Find the total revenue in part a). q Vectors in Three-Space Determine the exact magnitude of each vector. a) AB, joining A(2, 7, 8) to B( 5, 9, 1) p Calculate the dot product of each pair of vectors. a) u [5, 2], v [ 6, 7] b) u 3i 2j, v 3i 7j c) u [3, 2], v [4, 6] b) PQ, joining P(0, 3, 6) to Q(4, 9, 7) 13. Given the vectors a [3, 7, 8], b [ 6, 3, 4], and c [2, 5, 7], evaluate each expression. a) 5a 4b 3c b) 5a c c) b (c a ) 420 MHR Calculus and Vectors Chapter 7

14. If u [6, 1, 8] is orthogonal to v [k, 4, 5], determine the value(s) of k. 7.5 The Cross Product and Its Properties 15. Determine u v for each pair of vectors.

Use an example to verify that a (b c ) a b a c for all vectors a, b, and c. 7.6 Applications of the Dot Product and Cross Product 18.

r 10 cm 80 b) In what direction does the torque vector point? f 200 N 19. Determine the projection, and its magnitude, of u [ 2, 5, 3] on v [4, 8, 9]. PROBLEM WRAP- UP CHAPTER Go to www.mcgrawhill.

gsp, an applet that shows how a car engine creates torque with the combustion of air and gasoline pushing a piston down and turning the crankshaft. This is called the power stroke.

4 14. If u [6, 1, 8] is orthogonal to v [k, 4, 5], determine the value(s) of k. 7.5 The Cross Product and Its Properties 15. Determine u v for each pair of vectors. a) v u 200 b) u [4, 1, 3], v [3, 7, 8] 16. Determine the area of the parallelogram defined by the vectors u [6, 8, 9] and v [3, 1, 2]. 17. Use an example to verify that a (b c ) a b a c for all vectors a, b, and c. 7.6 Applications of the Dot Product and Cross Product 18. A force of 200 N is applied to a wrench in a clockwise direction at 80 to the handle, 10 cm from the centre of the bolt. a) Calculate the magnitude of the torque. r 10 cm 80 b) In what direction does the torque vector point? f 200 N 19. Determine the projection, and its magnitude, of u [ 2, 5, 3] on v [4, 8, 9]. PROBLEM WRAP- UP CHAPTER Go to and follow the links to 7review. Download the file EngineTorque.gsp, an applet that shows how a car engine creates torque with the combustion of air and gasoline pushing a piston down and turning the crankshaft. This is called the power stroke. Gases are released on the up, or exhaust, stroke, but no torque is produced. This is a highly simplified model of an internal combustion engine. Engineers who design real engines must take more factors into account, such as the fuel being used, the way it burns in the cylinder and the transfer of force along the connecting rods. Open the applet and click on the Start/Stop Engine button. a) As the crankshaft turns, at what point(s) would the torque be the greatest? Provide mathematical evidence. b) At what point(s) would the torque be the least? Provide mathematical evidence. c) Compare the rotation of the crankshaft to how you would use a wrench. Which method would be more efficient? Support your argument mathematically. Review MHR 421

5 Chapter 8 REVIEW 8.1 Equations of Lines in Two-Space and Three-Space 1. Write the vector and parametric equations of each line. a) m [1, 2], P( 3, 2) b) m [6, 5, 1], P( 9, 0, 4) c) parallel to the x-axis with z-intercept 7 d) perpendicular to the xy-plane and through (3, 0, 4) 2. Given each scalar equation, write a vector equation. a) 5x 2y 9 b) x 7y 10 c) x 8 d) x 4y 0 3. Write the scalar equation for each line. a) [x, y] [1, 4] t[2, 7] b) [x, y] [10, 3] t[5, 7] 4. A line is defined by the equation [x, y, z] [1, 1, 5] t[3, 4, 7]. a) Write the parametric equations for the line. b) Does the point (13, 15, 23) lie on the line? 5. The vertices of a parallelogram are the origin and points A( 1, 4), B(3, 6), and C(7, 2). Write the vector equations of the lines that make up the sides of the parallelogram. 6. A line has the same x-intercept as [x, y, z] [ 21, 8, 14] t[ 12, 4, 7] and the same y-intercept as [x, y, z] [6, 8, 12] s[2, 5, 4]. Write the parametric equations of the line. 8.2 Equations of Planes 7. Find three points on each plane. a) [x, y, z] [3, 4, 1] s[1, 1, 4] t[2, 5, 3] b) x 2y z 12 0 c) x 3k 4p y 5 2k p z 2 3k 2p 8. A plane contains the line [x, y, z] [2, 9, 10] t[3, 8, 7] and the point P(5, 1, 3). Write the vector and parametric equations of the plane. 9. Does P( 3, 4, 5) lie on each plane? a) [x, y, z] [1, 5, 6] s[2, 1, 3] t[1, 7, 1] b) 4x y 2z Do the points A(2, 1, 5), B( 1, 1, 10), and C(8, 5, 5) define a plane? Explain why or why not. 11. A plane is defined by the equation x 4y 2z 16. a) Find two vectors parallel to the plane. b) Determine the x-, y-, and z-intercepts. c) Write the vector and parametric equations of the plane. 8.3 Properties of Planes 12. Write the scalar equation of the plane with n [1, 2, 9] that contains P(3, 4, 0). 13. Write the scalar equation of this plane. [x, y, z] [5, 4, 7] s[0, 1, 0] t[0, 0, 1] 14. Write the scalar equation of each plane. a) parallel to the yz-plane with x-intercept 4 b) parallel to the vector a [3, 7, 1] and to the y-axis, and through (1, 2, 4) 8.4 Intersections of Lines in Two-Space and Three-Space 15. Determine the number of solutions for each linear system in two-space. If possible, solve each system. a) 2x 5y 6 x 9 7t y 4 3t b) [x, y] [9, 4] s[1,1] [x, y] [0, 9] t[3, 4] 502 MHR Calculus and Vectors Chapter 8

16. Write two other equations that have the same solution as this system of equations. 3x 4y 14 x 3y 18 17. Determine if the lines in each pair intersect.

Find the distance between these skew lines. [x, y, z] [1, 0, 1] s[2, 3, 4] [x, y, z] [8, 1, 3] t[4, 5, 1] 8.5 Intersections of Lines and Planes 19. Determine if each line intersects the plane.

2. 8.6 Intersections of Planes 21. Find the line of intersection for these two planes. 3x y z 10 5x 4y 2z 31 22. How do the planes in each system intersect?

a) 2x 5y 3z 0 x 3y 6z 19 3x 2y 9z 7 b) 8x 20y 16z 3 3x 15y 12z 10 2x 5y 4z 2 PROBLEM WRAP-UP CHAPTER When 3-D artists create objects that will eventually be animated, they start with a wire frame

6 16. Write two other equations that have the same solution as this system of equations. 3x 4y 14 x 3y Determine if the lines in each pair intersect. If so, find the coordinates of the point of intersection. a) [x, y, z] [1, 5, 2] s[1, 7, 3] [x, y, z] [ 3, 23, 10] t[1, 7, 3] b) [x, y, z] [15, 2, 1] s[4, 1, 1] [x, y, z] [13, 5, 4] t[ 5, 2, 3] 18. Find the distance between these skew lines. [x, y, z] [1, 0, 1] s[2, 3, 4] [x, y, z] [8, 1, 3] t[4, 5, 1] 8.5 Intersections of Lines and Planes 19. Determine if each line intersects the plane. If so, state the solution. a) 5x 2y 4z 23 [x, y, z] [ 17, 7, 6] t[4, 1, 3] b) x 4y 3z 11 [x, y, z] [ 1, 9, 16] t[3, 3, 5] 20. Find the distance between point P(3, 2, 0) and the plane 4x y 8z Intersections of Planes 21. Find the line of intersection for these two planes. 3x y z 10 5x 4y 2z How do the planes in each system intersect? a) 2x 5y 2z 3 x 2y 3z 11 2x y 5z 8 b) x 3y 2z 10 3x 5y z 1 6x 4y 7z 5 c) x 3y z 2 3x y z 14 5x 7y z Use the normal vectors of the planes to describe each system. a) 2x 5y 3z 0 x 3y 6z 19 3x 2y 9z 7 b) 8x 20y 16z 3 3x 15y 12z 10 2x 5y 4z 2 PROBLEM WRAP-UP CHAPTER When 3-D artists create objects that will eventually be animated, they start with a wire frame model of the object. Programmers then work to develop shaders that will be applied as materials for the models. Meanwhile, programmers and specialized technical artists create rigs, which the animator will use to control the motion of the models. A vehicle for your game is modelled with the wire frame shown in the diagram. Write the equations for all of the outside surfaces and edges m z 1.25 m y 2 m 0.75 m 0.5 m 2.25 m x Review MHR 503

5. A triangle has sides represented by the vectors (1, 2) and (5, 6). Determine the vector representing the third side.

5. A triangle has sides represented by the vectors (1, 2) and (5, 6). Determine the vector representing the third side. Vectors EXAM review Problem 1 = 8 and = 1 a) Find the net force, assume that points North, and points East b) Find the equilibrant force 2 = 15, = 7, and the angle between and is 60 What is the magnitude