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### 2 vectors

1. 1. VECTORS THEORIES
2. 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. 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
4. 4. Introduction: scalar and vector quantities Vectors <ul><li>A scalar quantity is defined completely by a single number with appropriate units </li></ul><ul><li>A vector quantity is defined completely when we know not only its magnitude (with units) but also the direction in which it operates </li></ul>Physical quantities can be divided into two main groups, scalar quantities and vector quantities.
5. 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
6. 6. Vector representation Vectors <ul><li>A vector quantity can be represented graphically by a line, drawn so that: </li></ul><ul><li>The length of the line denotes the magnitude of the quantity </li></ul><ul><li>The direction of the line (indicated by an arrowhead) denotes the direction in which the vector quantity acts. </li></ul>The vector quantity AB is referred to as or a
7. 7. Vector representation Two equal vectors Types of vectors Addition of vectors The sum of a number of vectors Vectors
8. 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. 9. Vector representation Vectors If two vectors, a and b , have the same magnitude but opposite direction then a = − b
10. 10. Vector representation Types of vectors Vectors <ul><li>A position vector occurs when the point A is fixed </li></ul><ul><li>A line vector is such that it can slide along its line of action </li></ul><ul><li>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 </li></ul>
11. 11. Vector representation Addition of vectors Vectors The sum of two vectors and is defined as the single vector
12. 12. Vector representation The sum of a number of vectors Programme 6: Vectors Draw the vectors as a chain.
13. 13. Vector representation The sum of a number of vectors Vectors If the ends of the chain coincide the sum is 0 .
14. 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. 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. 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. 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. 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. 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. 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. 21. Direction cosines Vectors Since:
22. 22. Direction cosines Vectors Defining: then: where [ l , m , n ] are called the direction cosines.
23. 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. 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. 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. 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. 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. 28. Vector product of two vectors Vectors If is a unit vector in the direction of: then: Notice that:
29. 29. Vector product of two vectors Vectors Since the coordinate vectors are mutually perpendicular: and
30. 30. Vector product of two vectors Vectors So, given: then: That is:
31. 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. 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. 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. 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.
35. 35. Learning outcomes <ul><li>Define a vector </li></ul><ul><li>Represent a vector by a directed straight line </li></ul><ul><li>Add vectors </li></ul><ul><li>Write a vector in terms of component vectors </li></ul><ul><li>Write a vector in terms of component unit vectors </li></ul><ul><li>Set up a system for representing vectors </li></ul><ul><li>Obtain the direction cosines of a vector </li></ul><ul><li>Calculate the scalar product of two vectors </li></ul><ul><li>Calculate the vector product of two vectors </li></ul><ul><li>Determine the angle between two vectors </li></ul><ul><li>Evaluate the direction ratios of a vector </li></ul>Vectors
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