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Scalars and vectors engineering physics.pptx
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- 3. Length of acar is 4.5 m physical quantity magnitude
- 4. Mass of goldbar is 1 kg physical quantity magnitude
- 5. physical quantity magnitude Timeis 12.76 s
- 6. Temperature is 36.8C physical quantity magnitude
- 7. A scalar isa physical quantity that has only a magnitude. Examples: • Mass • Length • Time • Temperature • Volume • Density
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- 9. California North Carolina Position ofCalifornia from North Carolina is 3600 km in west physical quantity magnitude direction
- 10. Displacement from USAto China is 11600 km in east physical quantity magnitude direction USA China
- 11. A vector isa physical quantity that has both a magnitude and a direction. Examples: • Position • Displacement • Velocity • Acceleration • Momentum • Force
- 12. Representation of avector Symbolically it is represented as
- 13. Representation of avector They are also represented by a single capital letter with an arrow above it. ⃗ P ⃗ B ⃗ A
- 14. Representation of avector Some vector quantities are represented by their respective symbols with an arrow above it. ⃗ F ⃗ v ⃗ r Position velocity Force
- 15. Types of Vectors (on thebasis of orientation)
- 16. Parallel Vectors Two vectorsare said to be parallel vectors, if they have same direction. ⃗ A ⃗ P ⃗ Q ⃗ B
- 17. Equal Vectors Two parallelvectors are said to be equal vectors, if they have same magnitude. ⃗ A ⃗ B ⃗ P ⃗ Q ⃗ A=⃗ B ⃗ P=⃗ Q
- 18. Anti-parallel Vectors Two vectorsare said to be anti-parallel vectors, if they are in opposite directions. ⃗ A ⃗ P ⃗ Q ⃗ B
- 19. Negative Vectors Two anti-parallelvectors are said to be negative vectors, if they have same magnitude. ⃗ A ⃗ B ⃗ P ⃗ Q ⃗ A=−⃗ B ⃗ P=−⃗ Q
- 20. Collinear Vectors Two vectorsare said to be collinear vectors, if they act along a same line. ⃗ A ⃗ B ⃗ P ⃗ Q
- 21. Types of Vectors (on thebasis of effect)
- 22. Polar Vectors Vectors havingstraight line effect are called polar vectors. Examples: • Displacement • Velocity • Acceleration • Force
- 23. Axial Vectors Vectors havingrotational effect are called axial vectors. Examples: • Angular momentum • Angular velocity • Angular acceleration • Torque
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- 26. Commutative Property ⃗ A ⃗ B ⃗ B ⃗ A ⃗ C ⃗ C Therefore, additionof vectors obey commutative law. = + = +
- 27. Unit vectors A unitvector is a vector that has a magnitude of exactly 1 and drawn in the direction of given vector. • It lacks both dimension and unit. • Its only purpose is to specify a direction in space. ⃗ A ^ 𝐴
- 28. • A givenvector can be expressed as a product of its magnitude and a unit vector. • For example may be represented as, ⃗ A= A ^ 𝐴 Unit vectors = magnitude of = unit vector along
- 29. Cartesian unit vectors 𝑥 𝑦 𝑧 ^ 𝑖 ^ 𝑗 ^ 𝑘 −𝑦 − 𝑥 −𝑧 - - -
- 30. Resolution of aVector It is the process of splitting a vector into two or more vectors in such a way that their combined effect is same as that of the given vector. 𝑡 𝑛 ⃗ A𝑡 ⃗ A𝑛 ⃗ A
- 31. Rectangular Components of2D Vectors ⃗ A ⃗ A𝑥 ⃗ A𝑦 O 𝑥 𝑦 ⃗ A ⃗ A=A𝑥 ^ 𝑖+ A𝑦 ^ 𝑗 θ θ A𝑥 ^ 𝑖 A𝑦 ^ 𝑗
- 32. A A𝑥 A𝑦 θ A𝑦 = Asinθ A𝑥= A cosθ ⇒ ⇒ Rectangular Components of 2D Vectors
- 33. Magnitude & directionfrom components ⃗ A=A𝑥 ^ 𝑖+ A𝑦 ^ 𝑗 A=√A𝑥 2 + A𝑦 2 Magnitude: Direction: θ=tan−1 (A 𝑦 A𝑥 ) θ A𝑥 A𝑦 A
- 34. Rectangular Components of3D Vectors 𝑥 𝑧 𝑦 ⃗ A ⃗ A𝑦 ⃗ A′ ⃗ A𝑧 ⃗ A𝑥 ⃗ A=⃗ A′ +⃗ A 𝑦 ⃗ A=⃗ A𝑥+⃗ A𝑧+⃗ A 𝑦 ⃗ A=⃗ A𝑥+⃗ A𝑦 +⃗ A𝑧 ⃗ A=A𝑥 ^ 𝑖+ A𝑦 ^ 𝑗+ A𝑧 ^ 𝑘
- 35. Rectangular Components of3D Vectors 𝑥 𝑧 𝑦 𝛼 A A𝑥 A𝑥= A cos𝛼
- 36. Rectangular Components of3D Vectors 𝑥 𝑧 𝑦 A A𝑦 𝛽 A𝑦 = Acos 𝛽
- 37. Rectangular Components of3D Vectors 𝑥 𝑧 𝑦 A A𝑧 𝛾 A𝑧= A cos𝛾
- 38. Magnitude & directionfrom components ⃗ A=A𝑥 ^ 𝑖+ A𝑦 ^ 𝑗+ A𝑧 ^ 𝑘 A=√A𝑥 2 +A𝑦 2 +A𝑧 2 Magnitude: Direction: 𝛼=cos−1 ( A𝑥 A ) 𝛽=cos − 1 (A 𝑦 A ) 𝛾=cos−1 ( A𝑧 A ) A A𝑧 𝛾 𝛽 𝛼 A𝑥 A𝑦
- 39. Adding vectors bycomponents ⃗ A=A𝑥 ^ 𝑖+ A𝑦 ^ 𝑗+ A𝑧 ^ 𝑘 ⃗ B=B𝑥 ^ 𝑖+B𝑦 ^ 𝑗+B𝑧 ^ 𝑘 Let us have t hen R𝑦=(A¿¿ 𝑦+B𝑦)¿
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- 41. Multiplying a vectorby a scalar • If we multiply a vector by a scalar s, we get a new vector. • Its magnitude is the product of the magnitude of and the absolute value of s. • Its direction is the direction of if s is positive but the opposite direction if s is negative.
- 42. Multiplying a vectorby a scalar If s is positive: ⃗ A 2⃗ A If s is negative: ⃗ A −3 ⃗ A
- 43. Multiplying a vectorby a vector • There are two ways to multiply a vector by a vector: • The first way produces a scalar quantity and called as scalar product (dot product). • The second way produces a vector quantity and called as vector product (cross product).
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- 45. Examples of scalarproduct W =⃗ F ∙ ⃗ s W = work done F = force s = displacement P=⃗ F ∙ ⃗ v P = power F = force v = velocity
- 46. Geometrical meaning ofScalar dot product A dot product can be regarded as the product of two quantities: 1. The magnitude of one of the vectors 2.The scalar component of the second vector along the direction of the first vector
- 47. Geometrical meaning ofScalar product θ A B cosθ θ A Acos θ B B
- 48. Properties of Scalarproduct 1 The scalar product is commutative.
- 49. Properties of Scalarproduct 2 The scalar product is distributive over addition. ⃗ A ∙ (⃗ B+⃗ C)=⃗ A ∙⃗ B+⃗ A ∙ ⃗ C
- 50. Properties of Scalarproduct 3 The scalar product of two perpendicular vectors is zero. ⃗ A ∙ ⃗ B=0
- 51. Properties of Scalarproduct 4 The scalar product of two parallel vectors is maximum positive. ⃗ A ∙⃗ B=A B
- 52. Properties of Scalarproduct 5 The scalar product of two anti-parallel vectors is maximum negative. ⃗ A ∙⃗ B=− A B
- 53. Properties of Scalarproduct 6 The scalar product of a vector with itself is equal to the square of its magnitude. ⃗ A ∙⃗ A= A2
- 54. Properties of Scalarproduct 7 The scalar product of two same unit vectors is one and two different unit vectors is zero. ^ 𝑖∙ ^ 𝑖= ^ 𝑗 ∙ ^ 𝑗=^ 𝑘∙ ^ 𝑘=(1)(1)cos 0 °=1 ^ 𝑖∙ ^ 𝑗= ^ 𝑗 ∙ ^ 𝑘= ^ 𝑘∙ ^ 𝑖=(1)(1)cos 90°=0
- 55. Calculating scalar productusing components ⃗ A=A𝑥 ^ 𝑖+ A𝑦 ^ 𝑗+ A𝑧 ^ 𝑘 ⃗ B=B𝑥 ^ 𝑖+B𝑦 ^ 𝑗+B𝑧 ^ 𝑘 Let us have t hen ⃗ A ∙⃗ B=A𝑥 B𝑥 + A𝑦 B𝑦 + A𝑧 B𝑧
- 56. Vector product ⃗ A ⃗ B θ ⃗ A ×⃗ B=ABsin θ ^ 𝑛=⃗ C
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- 58. Examples of vectorproduct ⃗ τ =⃗ r× ⃗ F = torque r = position F = force ⃗ L=⃗ r × ⃗ p ⃗ L=r psin θ ^ 𝑛 L = angular momentum r = position p = linear momentum
- 59. Geometrical meaning ofVector product θ A B B s i n θ θ A B A sin θ = Area of parallelogram made by two vectors
- 60. Properties of Vectorproduct 1 The vector product is anti-commutative.
- 61. Properties of Vectorproduct 2 The vector product is distributive over addition. ⃗ A ×(⃗ B+⃗ C)=⃗ A ×⃗ B+⃗ A ×⃗ C
- 62. Properties of Vectorproduct 3 The magnitude of the vector product of two perpendicular vectors is maximum. |⃗ A×⃗ B|=A B
- 63. Properties of Vectorproduct 4 The vector product of two parallel vectors is a null vector. ⃗ A × ⃗ B=⃗ 0
- 64. Properties of Vectorproduct 5 The vector product of two anti-parallel vectors is a null vector. ⃗ A × ⃗ B=⃗ 0
- 65. Properties of Vectorproduct 6 The vector product of a vector with itself is a null vector. ⃗ A × ⃗ A=⃗ 0
- 66. Properties of Vectorproduct 7 The vector product of two same unit vectors is a null vector. ^ 𝑖×^ 𝑖=^𝑗× ^ 𝑗=^ 𝑘×^ 𝑘 ¿(1)(1)sin0° ^ 𝑛=⃗ 0
- 67. Properties of Vectorproduct 8 The vector product of two different unit vectors is a third unit vector. ^ 𝑖 × ^ 𝑗=^ 𝑘 ^ 𝑗 × ^ 𝑘=^ 𝑖 ^ 𝑘× ^ 𝑖= ^ 𝑗 ^ 𝑗× ^ 𝑖=− ^ 𝑘 ^ 𝑘× ^ 𝑗=− ^ 𝑖 ^ 𝑖× ^ 𝑘=− ^ 𝑗
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- 69. Calculating vector productusing components ⃗ A=A𝑥 ^ 𝑖+ A𝑦 ^ 𝑗+ A𝑧 ^ 𝑘 ⃗ B=B𝑥 ^ 𝑖+B𝑦 ^ 𝑗+B𝑧 ^ 𝑘 Let us have t hen
- 70. Calculating vector productusing components ⃗ A × ⃗ B= | ^ 𝑖 ^ 𝑗 ^ 𝑘 A𝑥 A 𝑦 A𝑧 B𝑥 B𝑦 B𝑧 |
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