1. 5.0 VECTORS 5.2 Vectors in Two and Three Dimensions 5.2 Vectors in Two and Three Dimensions 5.3 Scalar Product 5.4 Vector Product 5.5 Application of Vectors in Geometry 5.1 Introduction to Vectors

2. 5.2 Vectors in Two Dimensions

3. Learning Outcomes (a) to understand the concept of vectors (b) to discuss the addition of vectors using the triangle laws (c) to define vectors in two dimension

4. Introduction to vectors Definition A scalar is a quantity that has only magnitude. Example: mass, temperature and volume A vector is a quantity that has both magnitude and direction. Example: velocity, force, weight and momentum

5. Vectors Representation Geometrically, vectors can be represented by a directed line segment. B A Head(Terminal Point) Tail (Initial Point)

6. A vector can be written as OR B A AB Magnitude ( also called norm ) is the length of the line AB Direction – The arrow head on the line AB

7. Types of Vectors Zero Vector (Null Vector) - A vector which has zero magnitude,

9. AND

11. Operation on Vectors Triangle Law (a) Addition of Vectors

12. a +b b + a Parallelogram Law

13. (b) Scalar Multiplication of Vectors

15. Vectors in Two Dimensions In xy-plane y x - a unit vector in the positive direction of y-axis - a unit vector in the positive direction of x-axis

16. Let A (3,4) y x A (3,4) C B O OB = 3, soOB = 3i BA = 4, soBA = 4j By using Triangle Law: Position Vector

17. i.e the position vector of A(3,4) is Similarly, the position vector of any point A(a 1,a 2 ) is Similarly, the magnitude of any vector

18. Direction Cosine Of A Vector α β A (3,4) y x C B O

19. Similarly, the direction cosine of any vector AND

20. Check

21. Addition and Subtraction of Vectors

22. NOTES

23. Scalar Multiplication

24. Example 1

25. y x B A O Solution: A ( 3, -1 ) B ( -2,3 )

26. Example 2 Given A (5,1) and B (2,-3). Find: (i) position vectors of A( ) and B( ) (ii)| + | and a unit vector of + (iii)Vector and its direction cosines

27. Solution:

28. Triangle Law Position vectors

30. CONCLUSION The position vector of any point P(a 1,a 2 ) is The magnitude of any vector

31. The direction cosine of any vectorAND Using position vectors