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Chapter 3. Vectors and Coordinate Systems

Chapter 3. Vectors and Coordinate Systems. Our universe has three dimensions, so some quantities also need a direction for a full description. For example, wind has both a speed and a direction; hence the motion of the wind is described by a vector.

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Chapter 3. Vectors and Coordinate Systems

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  1. Chapter 3. Vectors and Coordinate Systems Our universe has three dimensions, so some quantities also need a direction for a full description. For example, wind has both a speed and a direction; hence the motion of the wind is described by a vector. Chapter Goal: To learn how vectors are represented and used.

  2. Student Learning Objectives – Ch. 3 • To understand the basic properties of vectors. • To add and subtract vectors both graphically and using components. • To be able to decompose a vector into its components and to reassemble vector components into a magnitude and a direction. • To recognize and use the basic unit vectors. • To work with tilted coordinate systems.

  3. Graphical Vector Addition

  4. Tip to Tail Method

  5. Parallelogram Method

  6. Vector Addition Problem • Which figure shows A1 + A2 + A3?

  7. Which figure shows ?

  8. Multiplication by a scalar

  9. Vector Subtraction

  10. Vector Subtraction • Which figure shows 2A – B?

  11. Which figure shows 2 − ?

  12. Components of vectors

  13. Magnitude of A: A= (Ax2 + Ay2)1/2 Direction of A: θ = tan-1 (Ay/Ax)

  14. What are the x- and y-components Cx and Cy of vector ? Cx = 1 cm, Cy = –1 cm Cx= –3 cm, Cy = 1 cm Cx = –2 cm, Cy = 1 cm Cx= –4 cm, Cy = 2 cm Cx= –3 cm, Cy = –1 cm

  15. What are the x- and y-components Cx and Cy of vector ? Cx = 1 cm, Cy = –1 cm Cx= –3 cm, Cy = 1 cm Cx = –2 cm, Cy = 1 cm Cx= –4 cm, Cy = 2 cm Cx= –3 cm, Cy = –1 cm

  16. Workbook problems 12, 13, 15, 16, 18

  17. Workbook problems 12, 13, 15, 16, 18 - answers

  18. Workbook exercises 25-29

  19. Workbook exercises 25-29 - answers

  20. Tilted axes • Often is it convenient to tilt the coordinate axes (to represent an object on an incline for example). • The axes stay perpendicular to each other. • The unit vectors corespond to axes, not to “horizontal and vertical” so they are also tilted.

  21. Tilted axes • Cx = C cos θ • Cy = C sin θ • Note that θ is defined relative to the tilted x-axis and not to “horizontal”

  22. EXAMPLE 3.7 Finding the force perpendicular to a surface

  23. EXAMPLE 3.7 Finding the force perpendicular to a surface

  24. EXAMPLE 3.7 Finding the force perpendicular to a surface

  25. Workbook problems 26, 27,28,30, 31

  26. Chapter 3. Summary Slides

  27. Important Concepts

  28. Important Concepts

  29. Using Vectors

  30. Using Vectors

  31. Using Vectors

  32. Using Vectors

  33. Chapter 3. Clicker Questions

  34. Which figure shows ?

  35. Which figure shows 2 − ?

  36. Angle φthat specifies the direction of is given by tan–1(Cy /Cx) tan–1(Cx /|Cy|) tan–1(Cy /|Cx|) tan–1(Cx /Cy) tan–1(|Cx |/|Cy|)

  37. Angle φthat specifies the direction of is given by tan–1(Cy /Cx) tan–1(Cx /|Cy|) tan–1(Cy /|Cx|) tan–1(Cx /Cy) tan–1(|Cx |/|Cy|)

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