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Homework, Page 511

Homework, Page 511. 1. . Homework, Page 511. 5. . Homework, Page 511. 9. . Homework, Page 511. 13. . Homework, Page 511. 17. . Homework, Page 511. Find a unit vector in the direction of the given vector. 21. . Homework, Page 511.

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Homework, Page 511

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  1. Homework, Page 511 1.

  2. Homework, Page 511 5.

  3. Homework, Page 511 9.

  4. Homework, Page 511 13.

  5. Homework, Page 511 17.

  6. Homework, Page 511 Find a unit vector in the direction of the given vector. 21.

  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. Homework, Page 511 Find the component form of the vector. 29.

  9. Homework, Page 511 Find the magnitude and direction angle of the vector. 33.

  10. Homework, Page 511 Find the magnitude and direction angle of the vector. 37.

  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. 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. 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. 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. Homework, Page 511 57. Which of the following is the magnitude of the vector (2, –1)? A. 1 B. C. D. E. 5

  16. 6.2 Dot Product of Vectors

  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. Dot Product

  19. Properties of the Dot Product Let u, v, and w be vectors and let c be a scalar. • u·v=v·u • u·u=|u|2 • 0·u=0 • u·(v+w)=u·v+u·w(u+v) ·w=u·w+v·w • (cu) ·v=u·(cv)=c(u·v)

  20. Example Finding the Dot Product

  21. Angle Between Two Vectors

  22. Example Finding the Angle Between Vectors

  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. 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. Projection of u and v

  26. Example, Page 520 Find the projection of u onto v. Then write u as a sum of two orthogonal vectors, one of which is projvu. 26. u = (3, –7), v = (–2, –6)

  27. Work

  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. Homework • Homework Assignment #18 • Read Section 6.3 • Page 519, Exercises: 1 – 65 (EOO) • Quiz next time

  30. 6.3 Parametric Equations and Motion

  31. Quick Review

  32. Quick Review Solutions

  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. 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. Example Graphing Parametric Equations

  36. Example Graphing Parametric Equations

  37. Example Eliminating the Parameter

  38. Example Eliminating the Parameter

  39. Example Finding Parametric Equations for a Line

  40. Example Simulating Horizontal Motion

  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?

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