1. Chapter 3 : Vectors - Introduction - Addition of Vectors - Subtraction of Vectors - Scalar Multiplication of Vectors - Components of Vectors - Magnitude of Vectors - Product of 2 Vectors - Application of Scalar/Dot Product & Cross Product

2. Introduction Has magnitude (represent by length of arrow). direction (direction of the arrow either to the right, left, etc). Eg: move the brick 5m to the right VectorsScalars Has magnitude only. Eg: move the brick 5m.

3. Introduction Use an arrow connecting an initial point A to terminal point B. Denote Written as Magnitude of Vectors Representation

4. Introduction Vector in opposite direction,, but has same magnitude as. Vectors Negative

5. Introduction If we have 2 vectors, with same magnitude & direction. Equal Vectors

6. Addition of Vectors Any 2 vectors can be added by joining the initial point of to the terminal point of. Eg: 1. The Triangle Law

7. Addition of Vectors If 2 vector quantities are represented by 2 adjacent sides of a parallelogram, then the diagonal of parallelogram will be equal to the summation of these 2 vectors. Eg: The parallelogram law is affected by the triangle law. 2. The Parallelogram Law

8. Addition of Vectors The sum of a number of vectors

9. Subtraction of Vectors Is a special case of addition. Eg:

10. Scalar Multiplication k ; vector multiply with scalar, k.. Parallel Vectors

11. Scalar Multiplication

12. Components of Vectors – Unit Vectors

13. Vectors in 2 Dimensional (R 2 )

14. Vectors in 3 Dimensional (R 3 )

15. Exercise : Draw the vector

16. Components of Vectors

17. Magnitude of Vectors Exercise: Example: 1. For Any Vector

18. Magnitude of Vectors Example: 2. From one point to another point of vector

19. Magnitude of Vectors Solution:

20. Do Exercise 3.3 in Textbook page 70.

21. Unit Vectors Example:

22. Do Exercise 3.4 in Textbook page 70.

23. Direction Angles & Cosines

24. Example: Solution (i): Direction cosines Direction angles 90.77

25. Direction Angles & Cosines Solution (ii) Direction cosines Direction angles

26. Do Exercise 3.5 in Textbook page 72.

27. Do Tutorial 3 in Textbook page 85 : No. 2 (i) No. 3 (i) No. 4 No. 5 (iii) No. 6 (i)

28. Operations of Vectors by Components Example: Solution:

29. Do Exercise 3.6 in Textbook page 72.

30. Product of 2 Vectors Example: Solution: Dot Product / Scalar Product

31. Do Exercise 3.7 in Textbook page 73.

32. Find Angle Between 2 Vectors Example: Solution:

33. Do Exercise 3.8 in Textbook page 74.

34. Product of 2 Product Example: Cross Product / Vector Product

35. Product of 2 Product Cross Product / Vector Product Solution:

36. Do Exercise 3.9 in Textbook page 74.

37. Find Angle Between 2 Vectors

38. Applications of Vectors Projections The Area of Triangle & Parallelogram The Volume of Parallelepiped & Tetrahedron Equations of Planes Parametric Equations of Line in R 3 Distance from a Point to the Plane

39. i. Projections

40. Scalar projection of b onto a: Vector projection of b onto a:

41. Example : i.Given. Find the scalar projection and vector projection of b onto a ii.Find given that Solutions:

42. ii. The Area of Triangle and Parallelogram

43. Example : Solutions:

45. iii. The Volume of Parallelepiped and Tetrahedron A parallelepiped is a three-dimensional formed by six parallelogram. Define three vectors To represent the three edges that meet at one vertex. The volume of the parallelepiped is equal to the magnitude of their scalar triple product

46. Volume of Parallelepiped Volume of Tetrahedron

47. Example : Solution:

48. iv. Equations of Planes

49. Example: Solutions:

50. Example : Solutions:

51. v. Parametric Equations of a Line in

53. Parametric equations of a line : Cartesian equations :

54. Example : Solutions:

55. vi. Distance from a Point to the Plane

56. Example:

57. Solutions:

58. ii.