1. Chapter 1
Vectors

2. Vectors are arrows

3. What are these vectors?

4. Magnitude of a vector = Length of the arrow4 3

5. What are the magnitudes?

6. Magnitudes (solution)

7. Adding and subtracting vectors

8. Add and subtract

9. Solution

10. Notations

11. Vector Components 5 4 -3

12. Terminology

13. Decomposing a vector
Hint: Once you know one side of a right-angle triangle and one other angle, you can find all the lengths using cos, sin or tan.

14. A quick reminder

15. Trigonometry

16. Solution

17. Write down the following three vectors in i j notation
Write down the following three vectors in i j notation. Find the sum of these vectors also.
10o
4.5 5 4
50o
60o

18. Angle of a vector
Find the angles the four vectors make with the positive x-axis. y
30° x

19. Calculating the angles

20. Why is the shift needed?

21. (-1) times a vector? 5 4 3

22. 5 4 3 5 4 3

23. In General

24. Adding Vectors Diagrammatically
You are allowed to move an arrow around as long as you do not change its direction and length.
Method for adding vectors:
Move the arrows until the tail of one arrow is at the tip of the other arrow.
Trace out the resultant arrow.

25. Subtracting Vectors Diagrammatically

26. Example

27. Example

28. Adding vectors 1
Add the three vectors to find the total displacement.

29. Adding Vectors 2

30. Distance & Displacement
How far an object has traveled
Displacement (is a vector):
How far an object has traveled and in what direction

31. Distance or Displacement?
5 m, going East
Distance
Displacement
Distance is actually the magnitude of displacement

32. Addition of distance / displacement
= 4m + 3m = 7m
Displacement
= 5m, in the direction of the arrow

33. Another example
Distance = 2m + 4m + 2m + 4m = 12m
Displacement = 0m

34. Distance & Displacement

35. Scalar & Vector Scalars (e.g. distance, speed):
Quantities which are fully described by a magnitude alone.
Vectors (e.g. displacement, velocity):
Quantities which are fully described by both a magnitude and a direction.

36. Speed or Velocity?
5 m/s
5 m/s, going East
Speed
Velocity

37. Speed = | Velocity |
Speed can be interpreted as the magnitude of the velocity vector:

38. Summary 5 3 4 Three ways to represent a vector:
By an arrow in a diagram
By i, j components
By the magnitude and angle
You need to learn all! 5 4 3

39. Multiplying vectors Two different products:
Dot product (gives a scalar)
Cross product (gives a vector)

40. Math: Vector Dot Product (scalar product)

41. Dot Product Example

42. Dot Product Example

43. Vector cross product

44. Cross product example