1. STROUD Worked examples and exercises are in the text Programme 6: Vectors VECTORS PROGRAMME 6

2. STROUD Worked examples and exercises are in the text Programme 6: Vectors 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

3. STROUD Worked examples and exercises are in the text Programme 6: Vectors 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

4. STROUD Worked examples and exercises are in the text Programme 6: Vectors Introduction: scalar and vector quantities (a)A scalar quantity is defined completely by a single number with appropriate units (b)A vector quantity is defined completely when we know not only its magnitude (with units) but also the direction in which it operates Physical quantities can be divided into two main groups, scalar quantities and vector quantities.

5. STROUD Worked examples and exercises are in the text Programme 6: Vectors 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

6. STROUD Worked examples and exercises are in the text Programme 6: Vectors 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

7. STROUD Worked examples and exercises are in the text Programme 6: Vectors Vector representation A vector quantity can be represented graphically by a line, drawn so that: (a)The length of the line denotes the magnitude of the quantity (b)The direction of the line (indicated by an arrowhead) denotes the direction in which the vector quantity acts. The vector quantity AB is referred to as or a

8. STROUD Worked examples and exercises are in the text Programme 6: Vectors Vector representation Two equal vectors Types of vectors Addition of vectors The sum of a number of vectors

9. STROUD Worked examples and exercises are in the text Programme 6: Vectors Vector representation Two equal vectors If two vectors, a and b, are said to be equal, they have the same magnitude and the same direction

10. STROUD Worked examples and exercises are in the text Programme 6: Vectors Vector representation If two vectors, a and b, have the same magnitude but opposite direction then a = −b

11. STROUD Worked examples and exercises are in the text Programme 6: Vectors Vector representation Types of vectors (a)A position vector occurs when the point A is fixed (b)A free vector is not restricted in any way. It is completely defined by its length and direction and can be drawn as any one of a set of equal length parallel lines

12. STROUD Worked examples and exercises are in the text Programme 6: Vectors Vector representation Addition of vectors The sum of two vectors and is defined as the single vector

13. STROUD Worked examples and exercises are in the text Programme 6: Vectors Vector representation The sum of a number of vectors Draw the vectors as a chain.

14. STROUD Worked examples and exercises are in the text Programme 6: Vectors Vector representation The sum of a number of vectors If the ends of the chain coincide the sum is 0.

15. STROUD Worked examples and exercises are in the text Programme 6: Vectors 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

16. STROUD Worked examples and exercises are in the text Programme 6: Vectors 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

17. STROUD Worked examples and exercises are in the text Programme 6: Vectors Components of a given vector 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.

18. STROUD Worked examples and exercises are in the text Programme 6: Vectors Components of a given vector Components of a vector in terms of unit vectors The position vector, denoted by r can be defined by its two components in the Ox and Oy directions as: If we now define i and j to be unit vectors in the Ox and Oy directions respectively so that then:

19. STROUD Worked examples and exercises are in the text Programme 6: Vectors 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

20. STROUD Worked examples and exercises are in the text Programme 6: Vectors 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

21. STROUD Worked examples and exercises are in the text Programme 6: Vectors In three dimensions a vector can be defined in terms of its components in the three spatial direction Ox, Oy and Oz as: i is a unit vector in the Ox direction, j is a unit vector in the Oy direction and k is a unit vector in the Oz direction Vectors in space The magnitude of r can then be found from Pythagoras ’ theorem to be:

22. STROUD Worked examples and exercises are in the text Programme 6: Vectors 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

23. STROUD Worked examples and exercises are in the text Programme 6: Vectors 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

24. STROUD Worked examples and exercises are in the text Programme 6: Vectors Direction cosines The direction of a vector in three dimensions is determined by the angles which the vector makes with the three axes of reference:

25. STROUD Worked examples and exercises are in the text Programme 6: Vectors Direction cosines Since:

26. STROUD Worked examples and exercises are in the text Programme 6: Vectors Direction cosines Defining: then: where [l, m, n] are called the direction cosines.

27. STROUD Worked examples and exercises are in the text Programme 6: Vectors 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

28. STROUD Worked examples and exercises are in the text Programme 6: Vectors 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

29. STROUD Worked examples and exercises are in the text Programme 6: Vectors Scalar product of two 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 θ is the angle between them. The scalar product (dot product) is denoted by:

30. STROUD Worked examples and exercises are in the text Programme 6: Vectors Scalar product of two vectors If a and b are two parallel vectors, the scalar product of a and b is then: Therefore, given: then:

31. STROUD Worked examples and exercises are in the text Programme 6: Vectors 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

32. STROUD Worked examples and exercises are in the text Programme 6: Vectors 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

33. STROUD Worked examples and exercises are in the text Programme 6: Vectors Vector product of two vectors The vector product (cross product) of a and b, denoted by: is a vector with magnitude: and a direction perpendicular to both a and b such that a, b and form a right-handed set.

34. STROUD Worked examples and exercises are in the text Programme 6: Vectors Vector product of two vectors If is a unit vector in the direction of: then: Notice that:

35. STROUD Worked examples and exercises are in the text Programme 6: Vectors Vector product of two vectors Since the coordinate vectors are mutually perpendicular: and

36. STROUD Worked examples and exercises are in the text Programme 6: Vectors Vector product of two vectors So, given: then: That is:

37. STROUD Worked examples and exercises are in the text Programme 6: Vectors 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

38. STROUD Worked examples and exercises are in the text Programme 6: Vectors 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

39. STROUD Worked examples and exercises are in the text Programme 6: Vectors Angle between two 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

40. STROUD Worked examples and exercises are in the text Programme 6: Vectors 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

41. STROUD Worked examples and exercises are in the text Programme 6: Vectors 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

42. STROUD Worked examples and exercises are in the text Programme 6: Vectors Direction ratios 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.

43. STROUD Worked examples and exercises are in the text Programme 6: Vectors Learning outcomes Define a vector Represent a vector by a directed straight line Add vectors Write a vector in terms of component vectors Write a vector in terms of component unit vectors Set up a system for representing vectors Obtain the direction cosines of a vector Calculate the scalar product of two vectors Calculate the vector product of two vectors Determine the angle between two vectors Evaluate the direction ratios of a vector