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Chapter 3 Lecture

Chapter 3 Lecture. Chapter 3 Vectors and Coordinate Systems. Chapter Goal: To learn how vectors are represented and used. Slide 3-2. Chapter 3 Preview. Slide 3-3. Chapter 3 Preview. Slide 3-4. Chapter 3 Preview. Slide 3-5. Chapter 3 Reading Quiz. Slide 3-6. Reading Question 3.1.

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Chapter 3 Lecture

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  1. Chapter 3 Lecture

  2. Chapter 3 Vectors and Coordinate Systems Chapter Goal: To learn how vectors are represented and used. Slide 3-2

  3. Chapter 3 Preview Slide 3-3

  4. Chapter 3 Preview Slide 3-4

  5. Chapter 3 Preview Slide 3-5

  6. Chapter 3 Reading Quiz Slide 3-6

  7. Reading Question 3.1 What is a vector? • A quantity having both size and direction. • The rate of change of velocity. • A number defined by an angle and a magnitude. • The difference between initial and final displacement. • None of the above. Slide 3-7

  8. Reading Question 3.1 What is a vector? • A quantity having both size and direction. • The rate of change of velocity. • A number defined by an angle and a magnitude. • The difference between initial and final displacement. • None of the above. Slide 3-8

  9. Reading Question 3.2 What is the name of the quantity represented as ? • Eye-hat. • Invariant magnitude. • Integral of motion. • Unit vector in x-direction. • Length of the horizontal axis. Slide 3-9

  10. Reading Question 3.2 What is the name of the quantity represented as ? • Eye-hat. • Invariant magnitude. • Integral of motion. • Unit vector in x-direction. • Length of the horizontal axis. Slide 3-10

  11. Reading Question 3.3 This chapter shows how vectors can be added using • Graphical addition. • Algebraic addition. • Numerical addition. • Both A and B. • Both A and C. Slide 3-11

  12. Reading Question 3.3 This chapter shows how vectors can be added using • Graphical addition. • Algebraic addition. • Numerical addition. • Both A and B. • Both A and C. Slide 3-12

  13. Reading Question 3.4 To decompose a vector means • To break it into several smaller vectors. • To break it apart into scalars. • To break it into pieces parallel to the axes. • To place it at the origin. • This topic was not discussed in Chapter 3. Slide 3-13

  14. Reading Question 3.4 To decompose a vector means • To break it into several smaller vectors. • To break it apart into scalars. • To break it into pieces parallel to the axes. • To place it at the origin. • This topic was not discussed in Chapter 3. Slide 3-14

  15. Chapter 3 Content, Examples, and QuickCheck Questions Slide 3-15

  16. The velocity vector has both a magnitude and a direction. Vectors • A quantity that is fully described by a single number is called a scalar quantity (i.e., mass, temperature, volume). • A quantity having both a magnitude and a direction is called a vector quantity. • The geometric representation of a vector is an arrow with the tail of the arrow placed at the point where the measurement is made. • We label vectors by drawing a small arrow over the letter that represents the vector, i.e.,: for position, for velocity, for acceleration. Slide 3-16

  17. Properties of Vectors • Suppose Sam starts from his front door, takes a walk, and ends up 200 ft to the northeast of where he started. • We can write Sam’s displacement as: • The magnitude of Sam’s displacement is S |S|  200 ft, the distance between his initial and final points. Slide 3-17

  18. Properties of Vectors • Sam and Bill are neighbors. • They both walk 200 ft to the northeast of their own front doors. • Bill’s displacement B (200 ft, northeast) has the same magnitude and direction as Sam’s displacement S. • Two vectors are equal if they have the same magnitude and direction. • This is true regardless of the starting points of the vectors. • B  S. Slide 3-18

  19. Vector Addition • A hiker’s displacement is 4 miles to the east, then 3 miles to the north, as shown. • Vector is the net displacement: • Because and are at right angles, the magnitude of is given by the Pythagorean theorem: • To describe the direction of , we find the angle: • Altogether, the hiker’s net displacement is: Slide 3-19

  20. QuickCheck 3.1 Which of the vectors in the second row shows ? Slide 3-20

  21. QuickCheck 3.1 Which of the vectors in the second row shows ? Slide 3-21

  22. Example 3.1 Using Graphical Addition to Find a Displacement Slide 3-22

  23. Example 3.1 Using Graphical Addition to Find a Displacement Slide 3-23

  24. Parallelogram Rule for Vector Addition • It is often convenient to draw two vectors with their tails together, as shown in (a) below. • To evaluate FDE, you could move E over and use the tip-to-tail rule, as shown in (b) below. • Alternatively, FDE can be found as the diagonal of the parallelogram defined by D and E, as shown in (c) below. Slide 3-24

  25. Addition of More than Two Vectors • Vector addition is easily extended to more than two vectors. • The figure shows the path of a hiker moving from initial position 0 to position 1, then 2, 3, and finally arriving at position 4. • The four segments are described by displacement vectors D1, D2, D3 and D4. • The hiker’s net displacement, an arrow from position 0 to 4, is: • The vector sum is found by using the tip-to-tail method three times in succession. Slide 3-25

  26. More Vector Mathematics Slide 3-26

  27. QuickCheck 3.2 Which of the vectors in the second row shows 2 ? Slide 3-27

  28. QuickCheck 3.2 Which of the vectors in the second row shows 2 ? Slide 3-28

  29. Coordinate Systems and Vector Components • A coordinate system is an artificially imposed grid that you place on a problem. • You are free to choose: • Where to place the origin, and • How to orient the axes. • Below is a conventional xy-coordinate system and the four quadrants I through IV. The navigator had better know which way to go, and how far, if she and the crew are to make landfall at the expected location. Slide 3-29

  30. Component Vectors • The figure shows a vector Aand an xy-coordinate system that we’ve chosen. • We can define two new vectors parallel to the axes that we call the component vectors of A, such that: • We have broken A into two perpendicular vectors that are parallel to the coordinate axes. • This is called the decomposition of A into its component vectors. Slide 3-30

  31. Components • Suppose a vector A has been decomposed into component vectorsAx andAy parallel to the coordinate axes. • We can describe each component vector with a single number called the component. • The component tells us how big the component vector is, and, with its sign, which ends of the axis the component vector points toward. • Shown to the right are two examples of determining the components of a vector. Slide 3-31

  32. Tactics: Determining the Components of a Vector Slide 3-32

  33. QuickCheck 3.3 What are the x- and y-components of this vector? • 3, 2 • 2, 3 • –3, 2 • 2, –3 • –3, -2 Slide 3-33

  34. QuickCheck 3.3 What are the x- and y-components of this vector? • 3, 2 • 2, 3 • –3, 2 • 2, –3 • –3, -2 Slide 3-34

  35. QuickCheck 3.4 What are the x- and y-components of this vector? • 3, 4 • 4, 3 • –3, 4 • 4, –3 • –3, –4 Slide 3-35

  36. QuickCheck 3.4 What are the x- and y-components of this vector? • 3, 4 • 4, 3 • –3, 4 • 4, –3 • –3, –4 Slide 3-36

  37. QuickCheck 3.5 What are the x- and y-components of vector C? • 1, –3 • –3, 1 • 1, –1 • –4, 2 • 2, –4 Slide 3-37

  38. QuickCheck 3.5 What are the x- and y-components of vector C? • 1, –3 • –3, 1 • 1, –1 • –4, 2 • 2, –4 Slide 3-38

  39. Moving Between the Geometric Representation and the Component Representation • We will frequently need to decompose a vector into its components. • We will also need to “reassemble” a vector from its components. • The figure to the right shows how to move back and forth between the geometric and component representations of a vector. Slide 3-39

  40. Moving Between the Geometric Representation and the Component Representation • If a component vector points left (or down), you must manually insert a minus sign in front of the component, as done for By in the figure to the right. • The role of sines and cosines can be reversed, depending upon which angle is used to define the direction. • The angle used to define the direction is almost always between 0 and 90. Slide 3-40

  41. Example 3.3 Finding the Components of an Acceleration Vector Find the x- and y-components of the acceleration vector a shown below. Slide 3-41

  42. Example 3.3 Finding the Components of an Acceleration Vector Slide 3-42

  43. Example 3.4 Finding the Direction of Motion Slide 3-43

  44. Example 3.4 Finding the Direction of Motion Slide 3-44

  45. Unit Vectors • Each vector in the figure to the right has a magnitude of 1, no units, and is parallel to a coordinate axis. • A vector with these properties is called a unit vector. • These unit vectors have the special symbols: • Unit vectors establish the directions of the positive axes of the coordinate system. Slide 3-45

  46. Vector Algebra • When decomposing a vector, unit vectors provide a useful way to write component vectors: • The full decomposition of the vector Acan then be written: Slide 3-46

  47. QuickCheck 3.6 Vector C can be written –3î + ĵ. –4î + 2ĵ. C. î – 3ĵ. D. 2î – 4ĵ. E. î – ĵ. Slide 3-47

  48. QuickCheck 3.6 Vector C can be written –3î + ĵ. –4î + 2ĵ. C. î – 3ĵ. D. 2î – 4ĵ. E. î – ĵ. Slide 3-47

  49. Example 3.5 Run Rabbit Run! Slide 3-49

  50. Example 3.5 Run Rabbit Run! Slide 3-50

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