1. What is wrong with the following statement?
I live exactly 15 miles from here. Meet me there at 4 PM. I’ll cook dinner for you.

2. Vectors & Scalars
Measurements can be vectors or scalars.
vector = magnitude (size) & direction.
Scalar = magnitude (size) only.
Both have units.

3. List some measurements that might have a direction.
Velocity
Time
Acceleration
Force
mass

4. Some Vector & Scalars: Vectors (direction). Scalars (number unit)
Some Vector & Scalars: Vectors (direction) Scalars (number unit) no direction Displacement distance Velocity speed Acceleration temperature Force time Momentum mass
Displacement, d (s) and distance, d.

5. Delta means subtract Dx = xf – xi.
Displacement, d = change in position, Dx
Delta means subtract Dx = xf – xi.
Start at 2 cm end at 10 cm.
Dx = xf – xi.
d = 10 cm – 2 cm = 8 cm
Start at 10 cm end at 2 cm.
d = 2 cm – 10 cm = - 8 cm.
The + or – sign says which direction.
Same distance dif displacement.

6. Displacement (d)
Straight line distance with direction from starting point.
Distance is the path length and no direction.

7. Take a walk.

8. Representation of vectors: Vectors represented diagrammatically (graphically) by sketching scaled arrows.

9. 1. Sketch & label a vector arrow to represent 9 m/s left
1. Sketch & label a vector arrow to represent 9 m/s left. Use a scale of 1 cm = 1m/s
sketch 9 cm arrow pointing left
Label
9 m/s left
-9 m/s

10. arrowhead points in vector direction
Scaled vector arrows can be sketched onto graph axes to show direction:
scale labeled
arrowhead points in vector direction
Magnitude shown
by length of arrow
d = 8 m

11. Direction can be stated as compass directions or angles in degrees (azimuth). Zero degrees is to the right or east. Sketch into notes. Sketch the axes.

12. State this direction 3 different ways.
120o (azimuth)
30o W of N
60o N of W.

13. Positive & Negative Displacements direction from start point
Right
East
North Up
Left
West
South
Down

14. +30 m could be E, N, up, Right
- 25 could be W, S,
down,
Left
A negative sign negates the direction:
-40 km W
+40 km, E.

15. Ex: State the vector – 25 m, W as a positive.
+25 m, E
Or simply 25 m, E

16. Hwk Worksheet Intro Vectors.

17. Addition or Combining Vectors
Sometimes, more than one vector quantity are combined. In that case, we must combine (add or subtract) the individual vector components to find a resultant vector.

18. Examples of combining vector quantiti:
Walk west and turn and walk north.
Drive 20 km/h south then 50 km/h SE.

19. The resultant velocity depends on 2 pushes: the engine push & the wind push.

20. Combining Parallel Vectors at 0o or 180o.
Simply add then.
20 m right and 5m right.
+20 m + 5m = + 25 m or 25 m right.
Object is 25 m right of start point.
20 m left and 5m right.
- 20 m + 5m = - 15 m or 15 m left.
Object is 2 m left of start point.

21. We can also use scaled arrows sketched head to tail to represent components and find resultant vectors.

22. R = length from tail to tip of new arrow.
3. When vectors are at 0o or 180o (straight line). I walked 10 km, E, then 5 km, E. Find displacement. Sketch the component and the resultant displacement vector arrows with a scale 1 cm = 1 km .
0o Same direction
10 km, E +
5 km, E = 15 km, E = Resultant
R = length from tail to tip of new arrow.

23. 180o Opposite directions. I walked 10 km, E, then 5 km, W. 10 km, E -
5 km, W = 5 km, E = R
5 km, W
10 km, E
= 5 km, E = R

24. Combining (adding) Vectors not parallel:
Mathematical operations can be done on non-linear vectors – but not in the usual way. Their direction has to be taken into account. We cannot simply add 40 km North + 20 km North East. The resulting displacement is not 60 km.

25. Graphical analysis: a scaled diagram is used for any number of, & combination of vectors.
Methods:
1) Parallelogram (Two vectors only)
2) Tail to tip (head to tail). For any number of vectors to be added.

26. Parallelogram Method Football Vectors 3 min

27. Tail to Tip/Head to Tail Graphical Method to find Resultant
Sketch scaled arrows one after the other.

28. 1. Two people kick a ball at the same time or concurrently
1. Two people kick a ball at the same time or concurrently. One gives it a velocity of 6.5 m/s east, the other gives it a velocity of 4.5 m/s 30o N of E. What is the final resultant velocity? Both methods of solving.
4.5 m/s
30o
6.5 m/s

29. Tail to Tip: Sketch a diagram with a scale of 1cm = 1 m/s
Tail to Tip: Sketch a diagram with a scale of 1cm = 1 m/s. Sketch each vector separately 1 at a time. Place the tail of one of the vectors to the tip of the other.
4.5 cm
30o
30o
6.5 cm

30. Now connect a straight line, the resultant, from the tail of the unmoved (1st)vector to the tip of the 2nd vector (moved).
4.5 cm R
30o
6.5 cm

31. Measure the resultant with your ruler to get the magnitude
Measure the resultant with your ruler to get the magnitude. Measure the angle to get the direction of the resultant.
R=10.6 m/s
b=13o

32. Negative vectors are in the opposite direction of positive ones.
10 m/s East = +10 m/s West
36 km 20o N of E = +36 km 20o S of W.
What does –10 m South mean?
+10 m North

33. Subtraction: Just reverse the direction of the negative vector & add graphically (make your scaled diagram).
12 km East – 6 km south
12 km East + 6 km north.
13 m/s north – 5 m/s 20o N of E =
13 m/s north +5 m/s 20o S of W

34. Equilibrant is a vector that “neutralizes” the resultant
Equilibrant is a vector that “neutralizes” the resultant. It is equal and opposite the resultant. Ex: R = 25 m/s South, Equilibrant = 25 m/s North or (-25m/s S)

35. Wksht Prb Vector Sketching

36. Parallelogram Method
The parallelogram method is similar, but instead of moving a vector & sketching a triangle, you turn your two vector components into a parallelogram.

37. You may sketch both vectors from the origin
You may sketch both vectors from the origin. But you must turn the shape into a parallelogram

38. 4.5 km
40o
8.0 km
When the parallelogram is complete, sketch the resultant between the two original component vectors corner to corner. Measure the resultant and the new angle.

39. Try this:
A hiker walks 4.5 km at 40o. He then turns and walks 8 km due east. Find his resultant displacement and direction.
12 m/s, 14 o.

40. Note: the tail to tip (head to tail) vector diagram may be to resolve any components even more than two. The parallelogram method may be used to resolve only two vector components. The Pythagorean theorem may only be used for vectors at right angles.

41. Graphical Method Make a scale – plan the line lengths
Sketch graph axes
Sketch 1st vector to scale in appropriate direction from origin – place arrowhead at the end.
Sketch the 2nd vector in the appropriate direction to the tip (arrowhead) of the 1st vector.
Continue until all vectors sketched.
Connect the beginning (origin) to the tip of the last vector sketched w/ a straight line.
Measure the line (resultant) and convert the scale back to appropriate units. Use protractor to find direction.

42. Hwk read 3-1 do page 87 show scaled vector sketches.
Youtube Vector Diagram Lesson 13 minutes

43. Situations where No Diagrams Needed
Vectors along a line-add or subtract.
Vectors at right angles. Use Pythagorean Theorem.

44. If vectors at 0o or 180o to each other – simply add or subtract.

45. What is the resultant displacement for the following:
15 m North + 5 m North.
15 m North + 5 m South.
20 m, N
10 m, N

46. Vectors at right angles
a and b are the components,
and c is the resultant.

47. tan-1 0.5 = q q = 26.6o S of W R = 11 km tan q = 5 km 10 km
Ex 1: Find Resultant displacement & direction of R using Pythagorean:
10 km q
5 km
= R2
R = 11 km
tan = q
tan q = 5 km
10 km
q = 26.6o S of W

48. Ex 2: A dog walks 4 km to the east and then 9 km to the north
Ex 2: A dog walks 4 km to the east and then 9 km to the north. What is his resultant displacement?
9.8 km
66o N of E

49. Ex 3: A pirate in search of treasure follows a map and walks 45-m north, turns and walks 7.5-m east. Use trigonometry to find the displacement.
45.6-m, 9.5o E of North

50. Ex 4: Spiderman sprints forward for 115 m and then scales a vertical building straight up for 136 m. A. Make a rough sketch. B. Find his resultant displacement.

51. Read text section 3-2 do pg 91 #2 - 4 and pg 114 #22 - 24

52. Resolution of Resultant to Components
All 2-d vectors can be described as the sum of perpendicular vectors.
Instead of combining vector components to give resultant, we take resultant &resolve it (break it ) into perpendicular components.

53. A plane takes off at an angle
A plane takes off at an angle. Make a rough sketch of its velocity vector relative to the ground.
It is sometimes useful to break or “resolve” the vector into horizontal & vertical components.

54. Vector a can be broken down, or resolved into 2 perpendicular components: ay & ax.

55. ay is opposite the angle q, so. ay = a sin q
ay is opposite the angle q, so ay = a sin q . ax is adjacent to angle q, so ax = a cos q .

56. Need horizontal component of plane v. vx = v cos q.
Ex 1: How fast must a car be moving to stay beneath a plane taking off at 105 km/h at 25o to the ground?
Need horizontal component of plane v.
vx = v cos q.
vx = 105 km/h cos 25 = 95 km/h

57. What is the vertical plane velocity from the previous example?
vy = v sin q
105 km/h sin 25o
44 km/h

58. Ex 2: The landing speed of the space shuttle is 99. 7 m/s
Ex 2: The landing speed of the space shuttle is 99.7 m/s. If the shuttle is landing at an angle of 15o to the horizontal: a. How fast is it descending? b. What is its horizontal velocity?
a m/s
b m/s

59. Hwk Text pg 94 # 3 -7 show all work

60. Finding Resultant Algebraically
In what situation can we simply add or subtract vectors to find a resultant without making a diagram or using Pythagorean?
To find resultant of 2 or more vectors, we can resolve each vector into the X and Y components.

61. Find the resultant of the 2 vectors below:

62. Each vector can be resolved to X & Y components.

63. The X & Y components can be added:
To find the resultant.

64. Example Problem