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- 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
- 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. 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. 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. 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
- 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. 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.
- 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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