2. Find RS and the component form of RS. x. b) θ = 236, v = 35 y. b) 4i 3j c) 7( cos 200 i+ sin 200. a) 2u + v b) w 3v c) u 4v + 2w

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1 Pre Calculus Worksheet 6.1 For questions 1-3, let R = ( 5, 2) and S = (2, 8). 1. Sketch the vector RS and the standard position arrow for this vector. 2. Find RS and the component form of RS. 3. Show algebraicall that RS and the standard position vector are equivalent. 4. Use the given angle θ and magnitude to sketch the vector v in standard position and write the component form. a) θ = 55, v = 14 b) θ = 236, v = Find the magnitude and direction angle of the given vector. b) 4i 3j c) 7( cos 200 i+ sin 200 j ) a) 5,12 6. Given the vectors u = 1, 3, v = 2, 4 and w = 2, 5, perform the indicated operation. a) 2u + v b) w 3v c) u 4v + 2w

2 7. In a warehouse a bo is being pushed up a 15 inclined plane with a force of 2.5lb, as shown at the right. 2.5 a) Find the component form of the force. v 15 b) Give an interpretation of the horizontal and vertical components of the force. 8. Juana and Diego Garcia, ages si and four respectivel, own a strong and stubborn pupp named Corporal. It is so difficult to take Corporal for a walk that the have devised a scheme using two leashes. Suppose Juana pulls with a force of 23 lb at an angle of 18 up from the horizontal and Diego pulls with a force of 27 lb at an angle of 15 down from the horizontal. a) Make and label a sketch of this situation. b) How hard is Corporal pulling if the pupp holds the children at a stand-still? 9. A ship is heading due north at 12 mph. The current is flowing southwest at 4 mph. Find the actual bearing and speed of the ship. 10. An airplane is fling on a bearing of 340 at 325 mph. A wind is blowing with bearing 320 at 40 mph. a) Find the component form of the velocit of the airplane. b) Find the actual ground speed and direction of the airplane.

3 Pre Calculus Worksheet 6.2 In Eercises 1-4, find the dot product of u and v. 1. u = 5, 2, v = 8,13 2. u = 2,7, v = 5, 8 3. u= 2i 4 j, v= 8i+ 7j 4. u= 4i 11 j, v = 3j 5. Use a the dot product to find u. a) u = 8,15 b) u= 4i In Eercises 6-9, find the angle θ between the vectors. 6. u= 4, 3, v = 1,5 7. u= 2, 3, v = 3,5 8. u= 3i 3 jv, = 2i+ 2 3j 9. ( 3, 4) (8, 5) 10. Prove that the vectors u and v are orthogonal. u= 4, 1, v = 1, 4

4 11. Recall question 9 from worksheet 6.1, a ship is heading due north at 12 mph with a current is flowing southwest at 4 mph gave us the velocit of the ship as v s = 0,12 and the velocit of the current as v c = 2 2, a) Determine the angle between vectors vs and vc using the method taught in this lesson. b) Wh does the method taught in this lesson give us the same solution? Would finding the angle this wa make finding the bearing of the ship an easier? We have alread defined orthogonal vectors based on the dot product, but we can also use our knowledge of vectors to determine when two vectors are parallel. Two vectors are parallel when one vector is a constant multiple of the other vector. In other words, if a = kb where k is a constant, then vectors a and b are parallel. In eercises 12-14, determine whether the vectors u and v are parallel, orthogonal, or neither. Give a justification for our choice u= 5,4, v =, u= 15, 12, v = 4,5 14. u= 20, 6, v = 3, 10

5 15. Sketch a graph of an parallel and orthogonal vectors found in questions Does the graph confirm the answer ou originall predicted? Parallel: Orthogonal: 16. Find the work done lifting a 2600 pound car 5.5 feet. 17. Find the work done b lifting a 100 pound bag of potatoes 3 feet. 18. The angle between a 200 pound force, F, and AB = 2i+ 3j is 30. Find the work done b F in moving an object from A to B. Assume the distance from A to B is measured in feet. 19. The angle between a 75 pound force F and AB is 60, where A = ( 1,1) and B = (4,3). Find the work done b F in moving an object from A to B. Assume the distance from A to B is measured in feet.

6 Pre Calculus Worksheet 6.3 Da 1 No Graphing Calculator 1. Create a table of values to graph the parametric equations. a) = 2, t = t+ 5 b) 10 2 = t 2, = 3t For the parametric equations in question 1a) and 1b), eliminate the parameter and identif the graph of the parametric curve. Does this equation model what ou graphed in question 1? 3. Eliminate the parameter and identif the graph of the parametric curve. a) = 2 cos t, = 2sin t b) = 5 cos t, = 5sin t 4. Find a parameterization (pair of parametric equations) for a circle centered at the origin with a radius Suppose ou wanted to translate our circle in question 4 so that the center was at ( 5, 3). Write a new parameterization.

7 6. Graph each set of parametric equations on the graphing calculator in RADIAN MODE. Use t-min = 0, t-ma = 2π, t-step = 0.05 with the window: 6 6, 4 4. Sketch the curve and describe it with words. Use our sketches to answer question 7. HINT: Use a reciprocal to graph a secant. a) = 2 cos t b) = 4 cos t c) = 2sin t = 4sin t = 2sin t = 4 cos t d) = 2sec t e) = sec t f) = tan t = tan t = 2 tan t = 2sec t 7. a) How are the graphs 6a and 6b similar? How are the different? c) How does graph 6a compare to graph 6c? Be sure to watch when the calculator graphs them!! 8. Write the parametric equations from 6a, 6c, 6d and 6f as a Cartesian equation. In addition to helping us graph the conic sections, parametric equations (just like vectors) are useful because of their inherent horizontal and vertical components. We can use these components to write parametric equations for a line when given two points.

8 For eample: Consider the points A = (2, 4) and B = (5, 1). The vector AB is the directed line segment from A to B and it is a part of the line AB. Instead of thinking of the line in Cartesian units, aka slope and intercept, we can think of the line in terms of its horizontal component and vertical component like a vector. Step 1: Decide which point will be the initial point. Start at point A. Step 2: Determine the change in and the change in. Step 3: Write parametric equations with the format: = initial value + t = initial value + t =new coordinate old coordinate = 5 2= 3 (or right 3) =new coordinate old coordinate = 1 4= 5 (or down 5) = 2 + 3t = 4 5t Keep in mind that the parameter is t (and our independent variable) in both equations. Notice if we started with point B, we would have a different pair of parametric equations for the SAME curve because we can write man equivalent equations for the same line. 10. Find a parameterization (aka pair of parametric equations) for the line through the given points. a) ( 2,5) and ( 4, 2 ) b) ( 5,1 ) and ( 7,3) c) ( 6, 0) and ( 2,5) So, if we alread know how to write Cartesian equations for lines, wh do we need parametric equations? One ver good reason is the difference between lines and line segments. In question 10, we did not restrict the values we could use for t. It is eas to create a line segment (or part of an function) using a parameterization with a restricted interval for t, such as t 0 or 1 t TRUE or FALSE: The graph of the parametric equations = t 1, = 2t 1 for 1 t 3 is a line segment with endpoints (0, 1) and (2, 5). Justif our answer. 12. Find a parameterization for the line segment with endpoints ( 2,5) and ( 4, 2 ). 13. MULTIPLE CHOICE: Which of the following describes the graph of the parametric equations = 1 t, = 3t+ 2 for t 0? A. a line B. a line segment C. a parabola D. a circle E. a ra

9 Pre Calculus Worksheet 6.3 da 2 1. A beetle and an ant are walking along a straight line on a table from position A to B: Beetle at time t: A = ( 1, 5) and B = (1, 2) Ant at time t: A = (4, 3) and B = ( 3, 1) a. Does the beetle meet (and eat) the ant? The graph provided ma help. b. Setup two sets of parametric equations to model this situation. c. Graph the parametric equations ou wrote in b did our answer to part a change? Wh or wh not? d. What time would the beetle need to arrive at the ant s path in order to eat it? 2. Ben can sprint at the rate of 24 ft/sec and Jerr sprints at 20 ft/sec. Ben gives Jerr a 10 foot head start. The parametric equations below model the race; Ben is running in lane 3 and Jerr is running in lane 5. Ben = 20t = 3 Jerr 24t 10 = = 5 a) Use our calculator to simulate a 100 ard dash. List the viewing window used below. Be sure to set our mode to graph simultaneousl. b) Who is ahead after 3 seconds and b how much? c) Who wins the race? Solve graphicall.

10 3. Two opposing plaers in the game Capture the Flag are 100 feet part. On a signal, the both run to grab the flag that is on the ground halfwa between them. The faster runner, however, wasn t paing attention and hesitates 0.1 seconds. The following equations model the two plaers race to the flag. = 10( t 0.1) = 100 9t Plaer 1: 1 Plaer 2: 2 = 3 = 3 1 a) According to the equations above, which plaer is the faster runner and how do ou know? 2 b) Use our calculator to simulate the race. List our window. c) Who captures the flag and b how man feet? 4. Two cars enter the Datona 500, a 500-mile race conducted on a 2-mile oval. The first car averages 100 mph. The second car averages 140 mph, but is delaed 1 hour at the start of the race due to an electrical problem. a. Set up two sets of parametric equations (one set for each car) to model this situation. Hint: Since there is no vertical component, let = # to represent a lane for each car as we did in question 2. b. Use our graphing calculator to determine which car will finish first. When selecting the window, consider how long it will take each car to finish the race this is our parameter t. List the window used. c. Algebraicall, determine how long it takes for the second car to catch the first car. How man miles will be left in the race when this occurs? 5. A famine relief agenc drops food containers from an airplane to a famine area. The drop was made from an altitude of 1000 feet above ground level. a. Write an equation to model the height of the container (during free fall) as a function of time, t. b. Use parametric mode to simulate the drop during the first 6 seconds. After 4 seconds of free fall, the parachutes open. How man feet above ground are the food containers when the parachutes open?

11 6. Kirb hits a ball when it is 4 feet above the ground with an initial velocit of 120 ft/sec. The ball leaves the bat with a 40 angle of elevation and heads towards a fence. a) Write a pair of parametric equations for this situation. b) How far horizontall and verticall has the ball traveled after 2 seconds? c) If the ball is not caught and nothing obstructs its path, when will the ball hit the ground? d) What is the maimum height the ball reaches? When does the ball reach this maimum height? e) Determine (algebraicall and graphicall) whether the ball will clear the 35-ft fence that is 350 feet from home plate. 7. Ton and Sue are tossing lawn darts while standing 20 feet awa from the edge of the target on the ground. The radius of the circular target is 18 in. If Ton tosses the dart directl at the target and releases it 3 feet above the ground with at initial velocit of 30 ft/sec at a 70 a) Write a pair of parametric equations for this situation. b) Will Ton s dart hit the target?