1. © 2013 Pearson Education, Inc. Chapter Goal: To learn how vectors are represented and used. Chapter 3 Vectors and Coordinate Systems Slide 3-2

2. © 2013 Pearson Education, Inc. Chapter 3 Preview Slide 3-3

3. © 2013 Pearson Education, Inc. Chapter 3 Preview Slide 3-4

4. © 2013 Pearson Education, Inc. Chapter 3 Preview Slide 3-5

5. © 2013 Pearson Education, Inc.  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. Vectors Slide 3-16 The velocity vector has both a magnitude and a direction.

6. © 2013 Pearson Education, Inc.  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: Vector Addition Slide 3-19

7. © 2013 Pearson Education, Inc. Example 3.1 Using Graphical Addition to Find a Displacement Slide 3-22

8. © 2013 Pearson Education, Inc. Example 3.1 Using Graphical Addition to Find a Displacement Slide 3-23

9. © 2013 Pearson Education, Inc.  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. Parallelogram Rule for Vector Addition Slide 3-24

10. © 2013 Pearson Education, Inc.  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. Coordinate Systems and Vector Components Slide 3-29 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.

11. © 2013 Pearson Education, Inc.  The figure shows a vector A and 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. Component Vectors Slide 3-30

12. © 2013 Pearson Education, Inc.  If a component vector points left (or down), you must manually insert a minus sign in front of the component, as done for B y 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 . Moving Between the Geometric Representation and the Component Representation Slide 3-40

13. © 2013 Pearson Education, Inc. Find the x- and y- components of the acceleration vector a shown below. Example 3.3 Finding the Components of an Acceleration Vector Slide 3-41

14. © 2013 Pearson Education, Inc. Example 3.3 Finding the Components of an Acceleration Vector Slide 3-42

15. © 2013 Pearson Education, Inc. Example 3.4 Finding the Direction of Motion Slide 3-43

16. © 2013 Pearson Education, Inc. Example 3.4 Finding the Direction of Motion Slide 3-44

17. © 2013 Pearson Education, Inc.  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. Unit Vectors Slide 3-45

18. © 2013 Pearson Education, Inc.  When decomposing a vector, unit vectors provide a useful way to write component vectors:  The full decomposition of the vector A can then be written: Vector Algebra Slide 3-46

19. © 2013 Pearson Education, Inc. Example 3.5 Run Rabbit Run! Slide 3-49

20. © 2013 Pearson Education, Inc. Example 3.5 Run Rabbit Run! Slide 3-50

21. © 2013 Pearson Education, Inc.  We can perform vector addition by adding the x- and y- components separately.  This method is called algebraic addition.  For example, if D  A  B  C, then:  Similarly, to find R  P  Q we would compute:  To find T  cS, where c is a scalar, we would compute: Working With Vectors Slide 3-51

22. © 2013 Pearson Education, Inc. Example 3.6 Using Algebraic Addition to Find a Displacement Slide 3-52

23. © 2013 Pearson Education, Inc. Example 3.6 Using Algebraic Addition to Find a Displacement Slide 3-53

24. © 2013 Pearson Education, Inc. Example 3.6 Using Algebraic Addition to Find a Displacement Slide 3-54

25. © 2013 Pearson Education, Inc.  For some problems it is convenient to tilt the axes of the coordinate system.  The axes are still perpendicular to each other, but there is no requirement that the x- axis has to be horizontal.  Tilted axes are useful if you need to determine component vectors “parallel to” and “perpendicular to” an arbitrary line or surface. Tilted Axes and Arbitrary Directions Slide 3-55

26. © 2013 Pearson Education, Inc. The angle Φ that specifies the direction of vector is A.tan –1 (C x /C y ). B.tan –1 (C y /C x ). C.tan –1 (|C x |/C y ). D.tan –1 (|C x |/|C y |). E.tan –1 (|C y |/|C x |). QuickCheck 3.7 Slide 3-58

27. © 2013 Pearson Education, Inc. The angle Φ that specifies the direction of vector is A.tan –1 (C x /C y ). B.tan –1 (C y /C x ). C.tan –1 (|C x |/C y ). D.tan –1 (|C x |/|C y |). E.tan –1 (|C y |/|C x |). QuickCheck 3.7 Slide 3-59

28. © 2013 Pearson Education, Inc. Chapter 3 Summary Slides Slide 3-60

29. © 2013 Pearson Education, Inc. Important Concepts Slide 3-61

30. © 2013 Pearson Education, Inc. Important Concepts Slide 3-62