1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 1 Homework, Page 511 1.

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 2 Homework, Page 511 5.

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 3 Homework, Page 511 9.

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 4 Homework, Page 511 13.

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 5 Homework, Page 511 17.

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 6 Homework, Page 511 Find a unit vector in the direction of the given vector. 21.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 7 Homework, Page 511 Find a unit vector in the direction of the given vector. Write the answer in (a) component form and (b) as a linear combination of the standard unit vectors. 25.

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 8 Homework, Page 511 Find the component form of the vector. 29.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 9 Homework, Page 511 Find the magnitude and direction angle of the vector. 33.

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 10 Homework, Page 511 Find the magnitude and direction angle of the vector. 37.

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 11 Homework, Page 511 41.An airplane is flying on a bearing of 335º at 530 mph. Find the component form of the velocity of the airplane.

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 12 Homework, Page 511 45.A basketball is shot at a 70º angle with the horizontal direction with initial speed 10 fps. (a) Find the component form of the initial velocity. (b) Give an interpretation of the horizontal and vertical components of the velocity. The horizontal component is the horizontal speed, in the absence of air resistance, the ball will maintain. The vertical component is the initial vertical velocity of the ball, which will change continuously due to the force exerted by gravity on the ball.

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 13 Homework, Page 511 Find the direction and magnitude of the resultant force. 49.A force of 50 lb acts on an object at an angle of 45º. A second force of 75 lb acts on the object at an angle of –30º.

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 14 Homework, Page 511 53.A ship heads due south with the current flowing northwest. Two hours later the ship is 20 miles in the direction 30º west of south from the original starting point. Find the speed with no current of the ship and the rate of the current.

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 15 Homework, Page 511 57.Which of the following is the magnitude of the vector (2, –1)? A.1 B. C. D. E.5

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.2 Dot Product of Vectors

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 17 What you’ll learn about The Dot Product Angle Between Vectors Projecting One Vector onto Another Work … and why Vectors are used extensively in mathematics and science applications such as determining the net effect of several forces acting on an object and computing the work done by a force acting on an object.

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 18 Dot Product

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 19 Properties of the Dot Product Let u, v, and w be vectors and let c be a scalar. 1.u·v=v·u 2.u·u=|u| 2 3.0·u=0 4.u·(v+w)=u·v+u·w (u+v) ·w=u·w+v·w 5.(cu) ·v=u·(cv)=c(u·v)

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 20 Example Finding the Dot Product

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 21 Angle Between Two Vectors

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 22 Example Finding the Angle Between Vectors

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 23 Example Finding a Force to Overcome Gravitational Pull Suppose Rafaela is sitting on a sled on a 45º slope. If she and the sled have a combined weight of 140 lb, how much force must Juan apply to a rope tied to the sled to prevent its sliding down the hill?

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 24 Orthogonal Vectors Perpendicular vectors are sometimes referred to as orthogonal vectors. The vectors u and v are orthogonal if and only if u·v = 0.

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 25 Projection of u and v

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example, Page 520 Find the projection of u onto v. Then write u as a sum of two orthogonal vectors, one of which is proj v u. 26.u = (3, –7), v = (–2, –6) Slide 6- 26

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 27 Work

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 28 Example Finding the Work Done by a Constant Force Find the work done by a force F of 50 lb acting in the direction (2, 3) in moving an object five feet from (0, 0) to a point in the first quadrant along the line y = x.

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 29 Homework Homework Assignment #18 Read Section 6.3 Page 519, Exercises: 1 – 65 (EOO) Quiz next time

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.3 Parametric Equations and Motion

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 31 Quick Review

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 32 Quick Review Solutions

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 33 What you’ll learn about Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher … and why These topics can be used to model the path of an object such as a baseball or golf ball.

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 34 Parametric Curve, Parametric Equations The graph of the ordered pairs (x,y), where x = f(t) and y = g(t) are functions defined on an interval I of t-values, is a parametric curve. The equations are parametric equations for the curve, the variable t is a parameter, and I is the parameter interval.

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 35 Example Graphing Parametric Equations

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 36 Example Graphing Parametric Equations

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 37 Example Eliminating the Parameter

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 38 Example Eliminating the Parameter

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 39 Example Finding Parametric Equations for a Line

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 40 Example Simulating Horizontal Motion

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 41 Example Simulating Projectile Motion Matt hits a baseball that is 3 ft off the ground at an angle of 30° above the horizontal with an initial velocity of 125 fps. Does the ball clear a 20 ft fence 400 ft from the plate?