1. Vectors and scalars A scalar quantity can be described by a single number, with some meaningful unit 4 oranges 20 miles 5 miles/hour 10 Joules of energy 9 Volts

2. Vectors and scalars A scalar quantity can be described by a single number with some meaningful unit A vector quantity has a magnitude and a direction in space, as well as some meaningful unit. 5 miles/hour North 18 Newtons in the “x direction” 50 Volts/meter down

3. 3-1 Vectors and Their Components The simplest example is a displacement vector If a particle changes position from A to B, we represent this by a vector arrow pointing from A to B Figure 3-1 In (a) we see that all three arrows have the same magnitude and direction: they are identical displacement vectors. In (b) we see that all three paths correspond to the same displacement vector. The vector tells us nothing about the actual path that was taken between A and B. © 2014 John Wiley & Sons, Inc. All rights reserved.

4. Vectors and scalars A scalar quantity can be described by a single number with some meaningful unit A vector quantity has a magnitude and a direction in space, as well as some meaningful unit. To establish the direction, you MUST first have a coordinate system! Standard x-y Cartesian coordinates common Compass directions (N-E-S-W)

5. Drawing vectors Draw a vector as a line with an arrowhead at its tip. The length of the line shows the vector’s magnitude. The direction of the line shows the vector’s direction relative to a coordinate system (that should be indicated!) x y z 5 m/sec at 30 degrees from the x axis towards y in the xy plane

6. Drawing vectors Vectors can be identical in magnitude, direction, and units, but start from different places…

7. Drawing vectors Negative vectors refer to direction relative to some standard coordinate already established – not to magnitude.

8. Adding two vectors graphically Two vectors may be added graphically using either the head-to-tail method or the parallelogram method.

9. Adding two vectors graphically Two vectors may be added graphically using either the head-to-tail method or the parallelogram method.

10. Adding two vectors graphically

11. 3-1 Vectors and Their Components The vector sum, or resultant o Is the result of performing vector addition o Represents the net displacement of two or more displacement vectors o Can be added graphically as shown: Figure 3-2 Eq. (3-1) © 2014 John Wiley & Sons, Inc. All rights reserved.

12. 3-1 Vectors and Their Components Vector addition is commutative o We can add vectors in any order Eq. (3-2) Figure (3-3) © 2014 John Wiley & Sons, Inc. All rights reserved.

13. 3-1 Vectors and Their Components Vector addition is associative o We can group vector addition however we like Eq. (3-3) Figure (3-4) © 2014 John Wiley & Sons, Inc. All rights reserved.

14. 3-1 Vectors and Their Components A negative sign reverses vector direction We use this to define vector subtraction Eq. (3-4) Figure (3-5) Figure (3-6) © 2014 John Wiley & Sons, Inc. All rights reserved.

15. 3-1 Vectors and Their Components These rules hold for all vectors, whether they represent displacement, velocity, etc. Only vectors of the same kind can be added o (distance) + (distance) makes sense o (distance) + (velocity) does not © 2014 John Wiley & Sons, Inc. All rights reserved.

16. 3-1 Vectors and Their Components These rules hold for all vectors, whether they represent displacement, velocity, etc. Only vectors of the same kind can be added o (distance) + (distance) makes sense o (distance) + (velocity) does not Answer: (a) 3 m + 4 m = 7 m (b) 4 m - 3 m = 1 m © 2014 John Wiley & Sons, Inc. All rights reserved.

17. Adding more than two vectors graphically To add several vectors, use the head-to-tail method. The vectors can be added in any order.

18. Adding more than two vectors graphically—Figure 1.13 To add several vectors, use the head-to-tail method. The vectors can be added in any order.

19. Subtracting vectors Reverse direction, and add normally head-to-tail…

20. Subtracting vectors

21. Multiplying a vector by a scalar If c is a scalar, the product cA has magnitude |c|A. 

22. Addition of two vectors at right angles First add vectors graphically. Use trigonometry to find magnitude & direction of sum.

23. Addition of two vectors at right angles Displacement (D) = √(1.00 2 + 2.00 2 ) = 2.24 km Direction  = tan -1 (2.00/1.00) = 63.4º East of North

24. Note how the final answer has THREE things! Answer: 2.24 km at 63.4 degrees East of North Magnitude (with correct sig. figs!)

25. Note how the final answer has THREE things! Answer: 2.24 km at 63.4 degrees East of North Magnitude (with correct sig. figs!) Units

26. Note how the final answer has THREE things! Answer: 2.24 km at 63.4 degrees East of North Magnitude (with correct sig. figs!) Units Direction

27. Components of a vector Represent any vector by an x-component A x and a y-component A y. Use trigonometry to find the components of a vector: A x = Acos θ and A y = Asin θ, where θ is measured from the +x-axis toward the +y-axis.

28. Positive and negative components The components of a vector can be positive or negative numbers.

29. Finding components We can calculate the components of a vector from its magnitude and direction.

30. 3-1 Vectors and Their Components Components in two dimensions can be found by: Where θ is the angle the vector makes with the positive x axis, and a is the vector length The length and angle can also be found if the components are known © 2014 John Wiley & Sons, Inc. All rights reserved.

31. 3-1 Vectors and Their Components In the three dimensional case we need more components to specify a vector o (a,θ,φ) or (a x,a y,a z ) © 2014 John Wiley & Sons, Inc. All rights reserved.

32. 3-1 Vectors and Their Components In the three dimensional case we need more components to specify a vector o (a,θ,φ) or (a x,a y,a z ) Answer: choices (c), (d), and (f) show the components properly arranged to form the vector © 2014 John Wiley & Sons, Inc. All rights reserved.

33. Calculations using components We can use the components of a vector to find its magnitude and direction: We can use the components of a set of vectors to find the components of their sum:

34. Adding vectors using their components

35. Unit vectors A unit vector has a magnitude of 1 with no units. The unit vector î points in the +x-direction, points in the +y- direction, and points in the +z-direction. Any vector can be expressed in terms of its components as A =A x î+ A y + A z. 

36. 3-2 Unit Vectors, Adding Vectors by Components A unit vector o Has magnitude 1 o Has a particular direction o Lacks both dimension and unit o Is labeled with a hat: ^ We use a right-handed coordinate system o Remains right-handed when rotated © 2014 John Wiley & Sons, Inc. All rights reserved.

37. 3-2 Unit Vectors, Adding Vectors by Components The quantities a x i and a y j are vector components The quantities a x and a y alone are scalar components Vectors can be added using components Eq. (3-9) → © 2014 John Wiley & Sons, Inc. All rights reserved.

38. 3-2 Unit Vectors, Adding Vectors by Components To subtract two vectors, we subtract components Unit Vectors, Adding Vectors by Components Eq. (3-13) © 2014 John Wiley & Sons, Inc. All rights reserved.

39. 3-2 Unit Vectors, Adding Vectors by Components To subtract two vectors, we subtract components Unit Vectors, Adding Vectors by Components Answer: (a) positive, positive (b) positive, negative (c) positive, positive Eq. (3-13) © 2014 John Wiley & Sons, Inc. All rights reserved.

40. 3-3 Multiplying Vectors Multiplying two vectors: the scalar product o Also called the dot product o Results in a scalar, where a and b are magnitudes and φ is the angle between the directions of the two vectors: The commutative law applies, and we can do the dot product in component form Eq. (3-20) Eq. (3-22) Eq. (3-23) © 2014 John Wiley & Sons, Inc. All rights reserved.

41. 3-3 Multiplying Vectors A dot product is: the product of the magnitude of one vector times the scalar component of the other vector in the direction of the first vector Eq. (3-21) Figure (3-18) Either projection of one vector onto the other can be used To multiply a vector by the projection, multiply the magnitudes © 2014 John Wiley & Sons, Inc. All rights reserved.

42. 3-3 Multiplying Vectors Answer: (a) 90 degrees (b) 0 degrees (c) 180 degrees © 2014 John Wiley & Sons, Inc. All rights reserved.

43. The scalar product The scalar product of two vectors (the “dot product”) is A · B = ABcos 

44. The scalar product The scalar product of two vectors (the “dot product”) is A · B = ABcos 

45. The scalar product The scalar product of two vectors (the “dot product”) is A · B = ABcos  Useful for Work (energy) required or released as force is applied over a distance (4A) Flux of Electric and Magnetic fields moving through surfaces and volumes in space (4B)

46. Calculating a scalar product By components, A · B = A x B x + A y B y + A z B z Example: A = 4.00 m @ 53.0°, B = 5.00 m @ 130°

47. Calculating a scalar product By components, A · B = A x B x + A y B y + A z B z Example: A = 4.00 m @ 53.0°, B = 5.00 m @ 130° A x = 4.00 cos 53 = 2.407 A y = 4.00 sin 53 = 3.195 B x = 5.00 cos 130 = -3.214 B y = 5.00 sin 130 = 3.830 A x B x + A y B y = 4.50 meters A · B = ABcos  cos(130-53) = 4.50 meters 2

48. The vector product The vector product (“cross product”) of two vectors has magnitude and the right- hand rule gives its direction.

49. 3-3 Cross Products o The cross product of two vectors with magnitudes a & b, separated by angle φ, produces a vector with magnitude: And a direction perpendicular to both original vectors Direction is determined by the right-hand rule Place vectors tail-to-tail, sweep fingers from the first to the second, and thumb points in the direction of the resultant vector Eq. (3-24) © 2014 John Wiley & Sons, Inc. All rights reserved.

50. 3-3 Multiplying Vectors The upper shows vector a cross vector b, the lower shows vector b cross vector a Figure (3-19) © 2014 John Wiley & Sons, Inc. All rights reserved.

51. The vector product The vector product (“cross product”) A x B of two vectors is a vector Magnitude = AB sin  Direction = orthogonal (perpendicular) to A and B, using the “Right Hand Rule” A B A x B x y z

52. 3-3 Multiplying Vectors The cross product is not commutative To evaluate, we distribute over components: Therefore, by expanding (3-26): Eq. (3-25) Eq. (3-26) Eq. (3-27) © 2014 John Wiley & Sons, Inc. All rights reserved.

53. 3-3 Multiplying Vectors © 2014 John Wiley & Sons, Inc. All rights reserved.

54. 3-3 Multiplying Vectors Answer: (a) 0 degrees (b) 90 degrees © 2014 John Wiley & Sons, Inc. All rights reserved.

55. The vector cross product The cross product of two vectors is A x B (with magnitude ABsin  Useful for Torque from a force applied at a distance away from an axle or axis of rotation (4A) Calculating dipole moments and forces from Magnetic Fields on moving charges (4B)

56. 3-1 Vectors and Their Components Angles may be measured in degrees or radians Recall that a full circle is 360˚, or 2π rad Know the three basic trigonometric functions Figure (3-11) © 2014 John Wiley & Sons, Inc. All rights reserved.

57. 3-2 Unit Vectors, Adding Vectors by Components Vectors are independent of the coordinate system used to measure them We can rotate the coordinate system, without rotating the vector, and the vector remains the same All such coordinate systems are equally valid Figure (3-15) Eq. (3-15) Eq. (3-14) © 2014 John Wiley & Sons, Inc. All rights reserved.

58. Scalars and Vectors Scalars have magnitude only Vectors have magnitude and direction Both have units! Adding Geometrically Obeys commutative and associative laws Unit Vector Notation We can write vectors in terms of unit vectors Vector Components Given by Related back by Eq. (3-2) Eq. (3-5) Eq. (3-7) 3 Summary Eq. (3-3) Eq. (3-6) © 2014 John Wiley & Sons, Inc. All rights reserved.

59. Adding by Components Add component-by-component Scalar Times a Vector Product is a new vector Magnitude is multiplied by scalar Direction is same or opposite Cross Product Produces a new vector in perpendicular direction Direction determined by right- hand rule Scalar Product Dot product Eqs. (3-10) - (3-12) Eq. (3-22) 3 Summary Eq. (3-20) Eq. (3-24) © 2014 John Wiley & Sons, Inc. All rights reserved.