1. 6.6 Vectors

2. Direction: 45⁰ Magnitude=5 5 45⁰
Vector: A vector is a quantity that has both magnitude and direction
It is customary to represent a vector by using an arrow
terminal point
A vector is drawn as a directed line segment, the arrow shows the direction. It does not mean it is a ray. 5
45⁰
initial point
Direction: 45⁰
Magnitude=5

3. Direction: 45⁰ Magnitude=5 5
Vector: A vector is a quantity that has both magnitude and direction
A vector has no position. These are all the same vector 5
Magnitude=5
Direction: 45⁰

5. Direction: 45⁰ Magnitude=5 5
The position vector has its initial point at the origin 5
Magnitude=5
Direction: 45⁰

6. Direction: 45⁰ Magnitude=5 5
The position vector has its initial point at the origin 5
Magnitude=5
Direction: 45⁰

7. This is the unit vector i

8. This is the vector 2i

9. This is the vector 3i

10. This is the unit vector j

11. This is the vector 2j

12. This is the vector 3j

13. The position vector has its initial point at the origin and can be represented as the sum of component vectors i and j

14. The position vector has its initial point at the origin and can be represented as the sum of component vectors i and j

15. This is the position vector for w=2i+3j
Since it is difficult to write vectors using bold print by hand we can also represent this vector as

16. This is the position vector for v=-2i-3j

17. Draw three representations of the vector u=i-2j on the graph, be sure to make one the position vector.

18. Draw three representations of the vector u=-2i+j on the graph, be sure to make one the position vector.

19. Vector v has initial point P and terminal point Q
Vector v has initial point P and terminal point Q. Write v in ai+bj form.
Subtract the coordinates of the terminal point from the coordinates of the initial point. P Q

20. Vector v has initial point P and terminal point Q
Vector v has initial point P and terminal point Q. Write v in ai+bj form. Q P

21. Vector v has initial point P and terminal point Q
Vector v has initial point P and terminal point Q. Write v in ai+bj form. Q P

22. Find the magnitude 3 -4

23. To find the magnitude of a vector-2 -5

24. To find the magnitude of a vector

25. Draw three representations of the vector w=3i - 2j
Write the vector v with initial point P (-6,1) and terminal point Q (-2,-3) in ai+bj form.
Find the magnitude of the following vectors.
u= -12i -5j w= -10j

26. P odd

27. You can multiply a vector by a scalar

28. You can multiply a vector by a scalar.
Find

29. Find 3w
-6u

30. You can add two vectors geometrically by placing the terminal point of one at the initial point of the other.
Find
The sum is represented by the resultant vector

31. You can add two vectors algebraically by adding the corresponding components of the two vectors.
Find

32. Add
algebraically and graphically

33. You can subtract vectors by adding the opposite of the second vector

34. You can subtract two vectors algebraically by subtracting the corresponding components of the two vectors

35. You can subtract vectors by adding the opposite of the second vector

36. You can subtract two vectors algebraically by subtracting the corresponding components of the two vectors

37. Notebook Link

38. Find Algebraically.

39. Find Geometrically

40. Find Geometrically

41. Show geometrically and algebraically
P 751 # Geometrically & Algebraically, #25 – 32Algebraically
Show geometrically and algebraically
Show algebraically

42. Find the vector v in ai+bj given the magnitude and the angle it makes with the positive x-axis.4 b a

43. Find the vector v in ai+bj given the magnitude and the angle it makes with the positive x-axis.3

44. Find the vector v in ai+bj given the magnitude and the angle it makes with the positive x-axis.3 a

45. Find the vector v in ai+bj given the magnitude and the angle it makes with the positive x-axis.2

46. Find the vector v in ai+bj given the magnitude and the angle it makes with the positive x-axis.4 b a

47. The jet stream is blowing at 40 mph and the direction N30°E
The jet stream is blowing at 40 mph and the direction N30°E. Express its velocity as a vector v in terms of i and j. N
40mph W E S

48. A child pulls a wagon with a force of 25lbs on the handle that makes an angle 40° with the ground. Express its velocity as a vector v in terms of i and j.
25lbs

49. Find the magnitude , to the nearest hundredth and the direction angle θ, to the nearest tenth of a degree . 10 6

50. Find the magnitude , to the nearest hundredth and the direction angle θ, to the nearest tenth of a degree . 7 -2

51. Find the magnitude , to the nearest hundredth and the direction angle θ, to the nearest tenth of a degree . 5
-10

52. Unit Vector : A vector whose magnitude is one.
In many applications of vectors it is helpful to find to find the unit vector with the same direction as a given vector. 1 3
The unit vector for w

53. 5 4 3 Find the unit vector having the same direction as vector v. 5/5
4/5 3
3/5

54. Find the unit vector having the same direction as vector v.4 -2

55. Find the unit vector having the same direction as vector v.1 -3

56. P – 45 odd, 47 – 50, 61-63,65,66

57. Airspeed Vector + Windspeed Vector = Groundspeed Vector
Compass Heading
True Bearingg
Airspeed vector
Airspeed – Speed relative to the air or speed if there was no wind.
Compass Heading – The direction the plane would travel if there was no wind or the direction the engine is pushing the plane. This is the direction of the airspeed vector.
Groundspeed – The speed of the plane relative to the ground or the speed of the plane with the wind.
True Bearing or True Course – The direction the plane flies with the wind pushing it or the direction relative to the ground. This is the direction of the groundspeed vector.
Wind Speed – Speed of the wind.

59. Directions and Bearings
The direction to a point is stated as the number of degrees east or west of north or south.
For example, the direction of A from O is N30ºE. B is N60ºW from O. C is S70ºE from O. D is S80ºW from O.
                                                             
Note:
N30ºE means the direction is 30º east of north.

60. N E W S N E W S N E W S N E W S

61. Two forces F1 and F2, of magnitude 10 and 30 pounds respectively, act on an object. The direction of F1 is N20°E and the direction of F2 is N65°E. Find the magnitude and direction of the resultant force.
20°
10lbs
70°
20°
65°
30lbs
65°
25°
20°
65°

62. Two forces F1 and F2, of magnitude 10 and 30 pounds respectively, act on an object. The direction of F1 is N20°E and the direction of F2 is N65°E. Find the magnitude and direction of the resultant force.
20°
65°
37.74
22.08
37.74 lbs.
30.61

63. Two F1 and F2 forces, of magnitude 30 and 60 pounds respectively, act on an object. The direction of F1 is N10°E and the direction of F2 is N60°E. Find the magnitude and direction of the resultant force.
10°
30lbs
80°
10°
60°
60lbs
60°
30°
10°
60°

64. Two F1 and F2 forces, of magnitude 30 and 60 pounds respectively, act on an object. The direction of F1 is N10°E and the direction of F2 is N60°E. Find the magnitude and direction of the resultant force.
82.54
59.54
51.96
82.54 lbs.

65. A pilot whose plane can maintain an airspeed of 350 miles/hr sets the compass at N 15° E and is blown off course by a 90 mph wind blowing due east . What is the groundspeed and the true bearing of the plane.
15°
350 mph
75° 90
350 mph
15°

66. Angle with the positive x-axis 61.89°
A pilot whose plane can maintain an airspeed of 350 miles/hr sets the compass at N 15° E and is blown off course by a 90 mph wind blowing due east . What is the groundspeed and the true bearing of the plane.
383.28
350 mph
338.07
15°
180.59
Groundspeed: mph
Angle with the positive x-axis 61.89°
The groundspeed of the plane is mph and the true bearing is N28°E.

67. A boat whose speed in still water is 20 mph is traveling across a river perpendicular to the 5 mph current. What is the actual speed the boat is traveling and at what angle is the boat blown off course.
5 mph
20 mph
20.62
The actual speed of the boat is mph and it is blown off course at an angle of 14°

68. P – 74, 83, 84

71. Wkst

72. An airplane pilot with an airspeed of 400 miles/hour orients her plane due Southwest
(45 degrees South of West). The plane encounters a 100 mile/hour wind blowing towards the east. Determine the groundspeed and the true bearing of the plane.
225°
45°
400 mph
100
225°
45°
400 mph
100

73. Angle with positive x-axis = 237.12°
An airplane pilot with an airspeed of 400 miles/hour orients her plane due Southwest
(45 degrees South of West). The plane encounters a 100 mile/hour wind blowing towards the east. Determine the groundspeed and the true bearing of the plane.
225°
237.12°
45°
32.88°
400 mph
336.79
100
Groundspeed: mph
Angle with positive x-axis = °
The groundspeed of the plane is mph the true bearing of the plane is S33°W