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

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

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

4. STROUD Worked examples and exercises are in the text Introduction: scalar and vector quantities Programme 6: Vectors (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 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 Programme 6: Vectors

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

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

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

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

10. STROUD Worked examples and exercises are in the text Vector representation Types of vectors Programme 6: Vectors (a)A position vector occurs when the point A is fixed (b)A line vector is such that it can slide along its line of action (c)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

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

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

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

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

15. STROUD Worked examples and exercises are in the text Components of a given vector Programme 6: 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. STROUD Worked examples and exercises are in the text Components of a given vector Components of a vector in terms of unit vectors Programme 6: 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:

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

18. STROUD Worked examples and exercises are in the text Vectors in space 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: where k is a unit vector in the Oz direction The magnitude of r can then be found from Pythagoras’ theorem to be:

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

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

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

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

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

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

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

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

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

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

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

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

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