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Understanding Vectors: Representation and Application

Explore how vectors are represented and applied, from graphical and algebraic addition to vector decomposition. Learn about scalar and vector quantities, vector properties, and vector addition rules. Dive into examples and quick-check questions to reinforce your understanding.

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Understanding Vectors: Representation and Application

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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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