Chapter 3 Vectors and Coordinate Systems

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

2. Chapter 3 Preview Slide 3-3

3. Chapter 3 Preview Slide 3-4

4. Chapter 3 Preview Slide 3-5

5. 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

6. 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

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

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

9. 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

10. 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

11. 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

12. 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

13. 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

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

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

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

17. 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

18. 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

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

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

21. Working With Vectors • 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: Slide 3-51

22. Example 3.6 Using Algebraic Addition to Find a Displacement Slide 3-52

23. Example 3.6 Using Algebraic Addition to Find a Displacement Slide 3-53

24. Example 3.6 Using Algebraic Addition to Find a Displacement Slide 3-54

25. Tilted Axes and Arbitrary Directions • 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. Slide 3-55

26. QuickCheck 3.7 The angle Φ that specifies the direction of vector is • tan–1(Cx/Cy). • tan–1(Cy/Cx). • tan–1(|Cx|/Cy). • tan–1(|Cx|/|Cy|). • tan–1(|Cy|/|Cx|). Slide 3-58

27. QuickCheck 3.7 The angle Φ that specifies the direction of vector is • tan–1(Cx/Cy). • tan–1(Cy/Cx). • tan–1(|Cx|/Cy). • tan–1(|Cx|/|Cy|). • tan–1(|Cy|/|Cx|). Slide 3-59

28. Chapter 3 Summary Slides Slide 3-60

29. Important Concepts Slide 3-61

30. Important Concepts Slide 3-62