1. AP Physics Chapter 3 Vector

2. AP Physics • Turn in Chapter 2 Homework, Worksheet, & Lab • Take quiz • Lecture • Q&A

3. Vector and Scalar • Vector: • Magnitude: How large, how fast, … • Direction: In what direction (moving or pointing) • Representation depends on frame of reference • Scalar: • Magnitude only • No direction • Representation does not depend on frame of reference

4. Examples of vector and scalar • Vectors: • Position, displacement, velocity, acceleration, force, momentum, … • Scalars: • Mass, temperature, distance, speed, energy, charge, …

5. Vector symbol • Vector: bold or an arrow on top • Typed: v and V or • Handwritten: • Scalar: regular • v or V • v stands for the magnitude of vector v.

6. Adding Vectors • Graphical • Head-to-Tail (Triangular) • Parallelogram • Analytical (by components)

7. b a Graphical representation of vector: Arrow • An arrow is used to graphically represent a vector. • The length of the arrow represents the magnitude of the vector. head tail • The direction of the arrow represents the direction of the vector. • When comparing the magnitudes of vectors, we ignore directions. • Vector a is smaller than vector b because a is shorter than b.

8. A B C Equivalent Vectors • Two vectors are identical and equivalent if they both have the same magnitude and are in the same direction. • They do not have to start from the same point. (Their tails don’t have to be at the same point.) • A, B and C are all equivalent vectors.

9. A -A -A Negative of Vector • Vector -A has the same magnitude as vector A but points in the opposite direction. • If vector A and B have the same magnitude but point in opposite directions, then A = -B, and B = -A

10. A B A+B B A Adding Vectors: Head-to-Tail • Head-to-Tail method: Example: A + B • Draw vector A • Draw vector B starting from the head of A • The vector drawn from the tail of A to the head of B is the sum of A + B. Make sure arrows are parallel and of same length.

11. B A A B+A B A+B B A A+B=B+A A + B How about B + A? What can we conclude?

12. B A C C B A A+B+C A+B+C Resultant vector: from tail of first to head of last.

13. A B A -B A-B=A+(-B) • Draw vector A A-B • Draw vector -B from head of A. • The vector drawn from the tail of A to head of –B is then A – B.

14. b c c b a a What Are the Relationships?

15. Magnitude of sum A + B = C A and B in same direction A and B in opposite direction A and B at some angle max. c min. c • |A – B|  C  A + B

16. A+B B A+B A B A Adding Vectors: Parallelogram • Draw the two vectors from the same point • Construct a parallelogram with these two vectors as two adjacent sides • The sum is the diagonal vector starting from the same tail point. Advantage: No need to measure length.

17. N 35o Example W E S • Vector a has a magnitude of 5.0 units and is directed east. Vector b is directed 35o west of north and has a magnitude of 4.0 units. Construct vector diagrams for calculating a + b and b – a. Estimate the magnitudes and directions of a + b and b – a from your diagram.

18. Solution N -a b b-a a+b W E 35o 66o 50o a S Using ruler and protractor, we find: a+b: 4.3 unit, 50o North of East b-a: 8.0 unit, 66o West of North

19. Drop perpendicular lines from the head of vector a to the coordinate axes, the components of vector a can be found: Vector Components y a ay  x ax •  is the angle between the vector and the +x axis. • ax and ay are scalars.

20. Finding components of a vector • Resolving the vector • Decomposing the vector

21. y  (for 3-D) ay a Vector magnitude and direction ax x • The magnitude and direction of a vector can be found if the components (ax and ay) are given:  is the angle from the +x axis to the vector.

22. Example • A ship sets out to sail to a point 120 km due north. Before the voyage, an unexpected storm blows the ship to a point 100 km due east of its starting point. How far, and in what direction, must it now sail to reach its original destination?

23. A ship sets out to sail to a point 120 km due north. Before the voyage, an unexpected storm blows the ship to a point 100 km due east of its starting point. How far, and in what direction, must it now sail to reach its original destination? Solution • It must sail 156 km at 39.8o West of North to reach its original destination. N  c a =120  E b = 100

24. Practice: What are the magnitude and direction of vector y 5  x 4

25. Practice: What are the components of a vector that has a magnitude of 12 units and makes an angle of 126o with the positive x direction? y 126o x

26. i or j or • k or Unit vectors • Unit vector: magnitude of exactly 1 and points in a particular direction. • x direction: • y direction: • z direction:

27. Vector components and expression • Any vector can be written in its components and the unit vectors:

28. Terminology • axi is the vector component of a. • ax is the (scalar) component of a.

29. Example • Express the following vector in component and unit vector form. y a=12.0 units ay =30o ax x

30. y ry b by ay bx a r ax rx x 2-D Adding Vectors by Components • When adding vectors by components, we add components in a direction separately from other components. 3-D Component form:

31. Example • The minute hand of a wall clock measures 10 cm from axis to tip. What is the displacement vector of its tip (a) from a quarter after the hour to half past, (b) in the next half hour, and (c) in the next hour? • • •

32. y C (0,10) x A (10, 0) B (0,-10) Solution or magnitude and direction form. rBC Try b) and c) O rAB

33. Two vectors are given by a = 4i – 3j + k and b = -i + j + 4k. Find: a + b a– b a vector c such that a – b + c = 0 Practice

34. Practice: 54-19The two vectors a and b in Fig. 3-29 have equal magnitudes of 10.0 m and the angels are 1 = 30o and 2 = 105o. Find the (a) x and (b) y components of their vector sum r, (c) the magnitude of r, and (d) the angle r makes with the positive direction of the x axis. y b b=105o+30o=135o r 2 a a 1 x

35. 54-19 (Continued)

36. Vector Multiplication • More: • Scalar (aka dot or inner) product: a  b • Vector (aka cross) product: a  b • We cannot write ab if a and b are vectors. • But we still can write 2a since 2 is a scalar.

37. : angle between vector and +x axis : angle between two vectors Scalar product: ab b a •  (phi) is the angle between vector a and b. •  is always between 0o and 180o. (0o    180o) • The scalar product is a scalar  It has no direction. • What if the two vectors are perpendicular to each other? 

38. b  a b  a Physical meaning of ab ba is the projection of b onto a. ba Also ab is the project of a onto b. ab

39. Properties of scalar product • ab = ba • ii = jj = kk = 1 • ij = ji = jk = kj = ki = ik = 0 • aa = a2 • ab = 0 if a  b 6.ab = axbx+ayby+azbz

40. Angle between two vectors When we know magnitudes and : When we know the components: Put together:

41. z • C y x Example:a. Determine the components and magnitude of r = a – b + c if a = 5.0i + 4.0j – 6.0k, b = -2.0i + 2.0j + 3.0k, and c = 4.0i + 3.0 j + 2.0k.b. Calculate the angle between r and the positive z axis. O 5 11 A -7 B

42. Another approach We are looking for the angle between r and any vector in the z direction. Let’s choose the unit vector in the z direction, k

43. B A c 5 3 b  4 a  Practice: 55-43For the vectors in Fig. 3-35, with a = 4, b =3, and c =5, calculate

44. -4 -3 Approach 2 y b c b x a c

45. c 5 3 b 4 a Approach 3

46. d = -c c e = -b ce 5 3 b 4 a da Approach 4

47. A  B A Vector Product: c = a  b • Magnitude of c is: • c is a vector, and it has a direction given by the right-hand-rule (RHR): • Place the vectors a and b so that their tails are at the same point. • Extend your right arm and fingers in the direction of a. • Rotate your hands along your arm so that you can flap your fingers toward b through the smaller angle between a and b. Then • Your outstretched thumb points in the direction of c.

48. Properties of cross product b  a = - (ab) ab is  a, anda b is  b a  a = 0

49. Practice:Three vectors are given by a = 3.0i + 3.0j – 2.0k, b = -1.0i –4.0j + 2.0k, and c = 2.0i + 2.0j + 1.0k. Find (a) a• (b  c), (b) a• (b + c), and (c) a  (b + c).

50. Pg52-53P (2)