1. VECTORS THEORIES

2. Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios Vectors

3. Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios Vectors

5. Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios Vectors

7. Vector representation Two equal vectors Types of vectors Addition of vectors The sum of a number of vectors Vectors

8. Vector representation Two equal vectors Vectors If two vectors, a and b , are said to be equal, they have the same magnitude and the same direction

9. Vector representation Vectors If two vectors, a and b , have the same magnitude but opposite direction then a = − b

11. Vector representation Addition of vectors Vectors The sum of two vectors and is defined as the single vector

12. Vector representation The sum of a number of vectors Programme 6: Vectors Draw the vectors as a chain.

13. Vector representation The sum of a number of vectors Vectors If the ends of the chain coincide the sum is 0 .

14. Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios Vectors

15. Components of a given vector Vectors Just as can be replaced by so any single vector can be replaced by any number of component vectors so long as the form a chain beginning at P and ending at T.

16. Components of a given vector Components of a vector in terms of unit vectors Vectors The position vector , denoted by r can be defined by its two components in the O x and O y directions as: If we now define i and j to be unit vectors in the O x and Oy directions respectively so that then:

17. Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios Vectors

18. Vectors in space Vectors In three dimensions a vector can be defined in terms of its components in the three spatial direction O x , O y and O z as: where k is a unit vector in the O z direction The magnitude of r can then be found from Pythagoras’ theorem to be:

19. Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios Vectors

20. Direction cosines Vectors The direction of a vector in three dimensions is determined by the angles which the vector makes with the three axes of reference:

21. Direction cosines Vectors Since:

22. Direction cosines Vectors Defining: then: where [ l , m , n ] are called the direction cosines.

23. Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios Vectors

24. Scalar product of two vectors Vectors If a and b are two vectors, the scalar product of a and b is defined to be the scalar (number): where a and b are the magnitudes of the vectors and  is the angle between them. The scalar product ( dot product ) is denoted by:

25. Scalar product of two vectors Vectors If a and b are two parallel vectors, the scalar product of a and b is then: Therefore, given: then:

26. Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios Vectors

27. Vector product of two vectors Vectors The vector product (cross product) of a and b , denoted by: is a vector with magnitude: and a direction such that a , b and form a right-handed set.

28. Vector product of two vectors Vectors If is a unit vector in the direction of: then: Notice that:

29. Vector product of two vectors Vectors Since the coordinate vectors are mutually perpendicular: and

30. Vector product of two vectors Vectors So, given: then: That is:

31. Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios Vectors

32. Angle between two vectors Vectors Let a have direction cosines [ l , m , n ] and b have direction cosines [ l ′ , m ′ , n ′ ] Let and be unit vectors parallel to a and b respectively. therefore

33. Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios Vectors

34. Direction ratios Vectors Since the components a , b and c are proportional to the direction cosines they are sometimes referred to as the direction ratios of the vector.