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4-scalarsvectors-161127184703.pptx
- 1. SCALARS & VECTORS Yadwinder singh Lect. Physics Gsss Gowara
- 3. Length of a car is 4.5 m physical quantity magnitude
- 4. Mass of gold bar is 1 kg physical quantity magnitude
- 5. physical quantity magnitude Time is 12.76 s
- 6. Temperature is 36.8 °C physical quantity magnitude
- 7. A scalar is a physical quantity that has only a magnitude. Examples: • Mass • Length • Time • Temperature • Volume • Density
- 9. California North Carolina Position of California from North Carolina is 3600 km in west physical quantity magnitude direction
- 10. Displacement from USA to China is 11600 km in east physical quantity magnitude direction USA China
- 11. A vector is a physical quantity that has both a magnitude and a direction. Examples: • Position • Displacement • Velocity • Acceleration • Momentum • Force
- 12. Representation of a vector Symbolically it is represented as AB
- 13. Representation of a vector They are also represented by a single capital letter with an arrow above it. P B A
- 14. Representation of a vector 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 the basis of orientation)
- 16. Parallel Vectors Two vectors are said to be parallel vectors, if they have same direction. A P Q B
- 17. Equal Vectors Two parallel vectors are said to be equal vectors, if they have same magnitude. A B P Q A = B P = Q
- 18. Anti-parallel Vectors Two vectors are said to be anti-parallel vectors, if they are in opposite directions. A P Q B
- 19. Negative Vectors Two anti-parallel vectors are said to be negative vectors, if they have same magnitude. A B P Q A = −B P = −Q
- 20. Collinear Vectors Two vectors are said to be collinear vectors, if they act along a same line. A B P Q
- 21. Co-initial Vectors Two or more vectors are said to be co-initial vectors, if they have common initial point. B A C D
- 22. Co-terminus Vectors Two or more vectors are said to be co-terminus vectors, if they have common terminal point. B A C D
- 23. Coplanar Vectors Three or more vectors are said to be coplanar vectors, if they lie in the same plane. A B D C
- 24. Non-coplanar Vectors Three or more vectors are said to be non-coplanar vectors, if they are distributed in space. B C A
- 25. Types of Vectors (on the basis of effect)
- 26. Polar Vectors Vectors having straight line effect are called polar vectors. Examples: • Displacement • Velocity • Acceleration • Force
- 27. Axial Vectors Vectors having rotational effect are called axial vectors. Examples: • Angular momentum • Angular velocity • Angular acceleration • Torque
- 30. Parallelogram Law A B A B A B B A C C=A+B
- 32. Commutative Property A B B A C C Therefore, addition of vectors obey commutative law. C=A+B =B +A
- 33. Associative Property B A C Therefore, addition of vectors obey associative law. D= (A+B)+C = A+(B +C) D B A C D
- 34. Subtraction of vectors B A −B A The subtraction of B from vectorA is defined as the addition of vector −B to vector A. A-B = A+(−B)
- 36. Magnitude of Resultant P O B A C Q R M θ θ cos θ = AM AC AM = AC cos θ ⇒ In∆CAM, OC2 = OM2 + CM2 OC2 = (OA + AM)2 + CM2 OC2 = OA2 + 2OA × AM + AM2 + CM2 OC2 = OA2 + 2OA × AM + AC2 R2 = P2 + 2P × Q cos θ + Q2 In∆OCM, R = P2 + 2PQ cos θ + Q2 OC2 = OA2 + 2OA × AC cos θ + AC2
- 37. Direction of Resultant P O A C Q R M α θ In∆CAM, cos θ = AM AC AM = AC cos θ ⇒ sin θ = CM AC CM = AC sin θ ⇒ In∆OCM, tan α = CM OM tan α = CM OA + AM tan α = AC sin θ OA + AC cos θ tan α = Q sin θ P + Q cos θ B
- 38. Case I – Vectors are parallel (𝛉 = 𝟎°) P Q R R = P2 + 2PQ cos 0° + Q2 R = P2 + 2PQ + Q2 R = (P + Q)2 + = Magnitude: Direction: R = P + Q tan α = Q sin 0° P + Q cos 0° tan α = 0 P + Q = 0 α = 0°
- 39. Case II – Vectors are perpendicular (𝛉 = 𝟗𝟎°) P Q R R = P2 + 2PQ cos 90° + Q2 R = P2 + 0 + Q2 + = Magnitude: Direction: tan α = Q sin 90° P + Q cos 90° = Q P + 0 α = tan−1 Q P P Q R = P2 + Q2 α
- 40. Case III – Vectors are anti-parallel (𝛉 = 𝟏𝟖𝟎°) P Q R R = P2 + 2PQ cos 180° + Q2 R = P2 − 2PQ + Q2 R = (P − Q)2 − = Magnitude: Direction: R = P − Q tan α = Q sin 180° P + Q cos 180° = 0 α = 0° If P > Q: If P < Q: α = 180°
- 41. Unit vectors A unit vector 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
- 42. • A given vector can be expressed as a product of its magnitude and a unit vector. • For example A may be represented as, A = A 𝐴 Unit vectors A= magnitude of A 𝐴= unit vector along A
- 44. Resolution of a Vector 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
- 45. Rectangular Components of 2D Vectors A A𝑥 A𝑦 O 𝑥 𝑦 A A = A𝑥𝑖 + A𝑦𝑗 θ θ A𝑥𝑖 A𝑦𝑗
- 46. A A𝑥 A𝑦 θ sin θ = A𝑦 A cos θ = A𝑥 A A𝑦 = A sin θ A𝑥 = A cos θ ⇒ ⇒ Rectangular Components of 2D Vectors
- 47. Magnitude & direction from components A = A𝑥𝑖 + A𝑦𝑗 A = A𝑥 2 + A𝑦 2 Magnitude: Direction: θ = tan−1 A𝑦 A𝑥 θ A𝑥 A𝑦 A
- 48. Rectangular Components of 3D Vectors 𝑥 𝑧 𝑦 A A𝑦 A′ A𝑧 A𝑥 A = A′ + A𝑦 A = A𝑥 + A𝑧 + A𝑦 A = A𝑥 + A𝑦 + A𝑧 A = A𝑥𝑖 + A𝑦𝑗 + A𝑧𝑘
- 49. Rectangular Components of 3D Vectors 𝑥 𝑧 𝑦 𝛼 A A𝑥 cos 𝛼 = A𝑥 A A𝑥 = A cos 𝛼
- 50. Rectangular Components of 3D Vectors 𝑥 𝑧 𝑦 A A𝑦 𝛽 cos 𝛽 = A𝑦 A A𝑦 = A cos 𝛽
- 51. Rectangular Components of 3D Vectors 𝑥 𝑧 𝑦 A A𝑧 𝛾 cos 𝛾 = A𝑧 A A𝑧 = A cos 𝛾
- 52. Magnitude & direction from components A = A𝑥𝑖 + A𝑦𝑗 + A𝑧𝑘 A = A𝑥 2 + A𝑦 2 + A𝑧 2 Magnitude: Direction: 𝛼 = cos−1 A𝑥 A 𝛽 = cos−1 A𝑦 A 𝛾 = cos−1 A𝑧 A A A𝑧 𝛾 𝛽 𝛼 A𝑥 A𝑦
- 53. Adding vectors by components A = A𝑥𝑖 + A𝑦𝑗 + A𝑧𝑘 B = B𝑥𝑖 + B𝑦𝑗 + B𝑧𝑘 Let us have then R = A + B R = A𝑥𝑖 + A𝑦𝑗 + A𝑧𝑘 + B𝑥𝑖 + B𝑦𝑗 + B𝑧𝑘 R = (A𝑥 + B𝑥)𝑖 + (A𝑦 + B𝑦)𝑗 + (A𝑧 + B𝑧)𝑘 R𝑥𝑖 + R𝑦𝑗 + R𝑧𝑘 = (A𝑥+B𝑥)𝑖 + (A𝑦+B𝑦)𝑗 + (A𝑧+B𝑧)𝑘 R𝑥 = (A𝑥 + B𝑥) R𝑦 = (A𝑦 + B𝑦) R𝑧 = (A𝑧 + B𝑧)
- 55. Multiplying a vector by a scalar • If we multiply a vector A by a scalar s, we get a new vector. • Its magnitude is the product of the magnitude of A and the absolute value of s. • Its direction is the direction of A if s is positive but the opposite direction if s is negative.
- 56. Multiplying a vector by a scalar If s is positive: A 2A If s is negative: A −3A
- 57. Multiplying a vector by 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).
- 58. Scalar product A B θ A ∙ B = AB cos θ
- 59. Examples of scalar product W = F ∙ s W = Fs cos θ W = work done F = force s = displacement P = F ∙ v P = Fv cos θ P = power F = force v = velocity
- 60. Geometrical meaning of Scalar 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
- 61. Geometrical meaning of Scalar product θ A ∙ B = A(B cos θ) A ∙ B = (A cos θ)B A θ A A cos θ B B
- 62. Properties of Scalar product 1 The scalar product is commutative. A ∙ B = AB cos θ B ∙ A = BA cos θ A ∙ B = B ∙ A
- 63. Properties of Scalar product 2 The scalar product is distributive over addition. A ∙ B + C = A ∙ B + A ∙ C
- 64. Properties of Scalar product 3 The scalar product of two perpendicular vectors is zero. A ∙ B = AB cos 90 ° A ∙ B = 0
- 65. Properties of Scalar product 4 The scalar product of two parallel vectors is maximum positive. A ∙ B = AB cos 0 ° A ∙ B = AB
- 66. Properties of Scalar product 5 The scalar product of two anti-parallel vectors is maximum negative. A ∙ B = AB cos 180 ° A ∙ B = −AB
- 67. Properties of Scalar product 6 The scalar product of a vector with itself is equal to the square of its magnitude. A ∙ A = AA cos 0 ° A ∙ A = A2
- 68. Properties of Scalar product 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
- 69. Calculating scalar product using components A = A𝑥𝑖 + A𝑦𝑗 + A𝑧𝑘 B = B𝑥𝑖 + B𝑦𝑗 + B𝑧𝑘 Let us have then A ∙ B = A𝑥𝑖 + A𝑦𝑗 + A𝑧𝑘 ∙ (B𝑥𝑖 + B𝑦𝑗 + B𝑧𝑘) A ∙ B = A𝑥𝑖 ∙ B𝑥𝑖 + B𝑦𝑗 + B𝑧𝑘 + A𝑦𝑗 ∙ B𝑥𝑖 + B𝑦𝑗 + B𝑧𝑘 + A𝑧𝑘 ∙ B𝑥𝑖 + B𝑦𝑗 + B𝑧𝑘 = A𝑥B𝑥𝑖 ∙ 𝑖 + A𝑥B𝑦𝑖 ∙ 𝑗 + A𝑥B𝑧𝑖 ∙ 𝑘 +A𝑦B𝑥𝑗 ∙ 𝑖 + A𝑦B𝑦𝑗 ∙ 𝑗 + A𝑦B𝑧𝑗 ∙ 𝑘 +A𝑧B𝑥𝑘 ∙ 𝑖 + A𝑧B𝑦𝑘 ∙ 𝑗 + A𝑧B𝑧𝑘 ∙ 𝑘 = A𝑥B𝑥(1) + A𝑥B𝑦(0) + A𝑥B𝑧(0) +A𝑦B𝑥(0) + A𝑦B𝑦(1) + A𝑦B𝑧(0) +A𝑧B𝑥(0) + A𝑧B𝑦(0) + A𝑧B𝑧(1) A ∙ B = A𝑥B𝑥 + A𝑦B𝑦 + A𝑧B𝑧
- 70. Vector product A B θ A × B = AB sin θ 𝑛 = C
- 72. Examples of vector product τ = r × F τ = rF sin θ 𝑛 τ = torque r = position F = force L = r × p L = rp sin θ 𝑛 L = angular momentum r = position p = linear momentum
- 73. Geometrical meaning of Vector product θ A × B = A(B sin θ) A B A × B = A sin θ B θ A B A sin θ A × B = Area of parallelogram made by two vectors
- 74. Properties of Vector product 1 The vector product is anti-commutative. A × B = AB sin θ 𝑛 B × A = BA sin θ (−𝑛) = −AB sin θ 𝑛 A × B ≠ B × A
- 75. Properties of Vector product 2 The vector product is distributive over addition. A × B + C = A × B + A × C
- 76. Properties of Vector product 3 The magnitude of the vector product of two perpendicular vectors is maximum. A × B = AB sin 90 ° A × B = AB
- 77. Properties of Vector product 4 The vector product of two parallel vectors is a null vector. A × B = AB sin 0 °𝑛 A × B = 0
- 78. Properties of Vector product 5 The vector product of two anti-parallel vectors is a null vector. A × B = AB sin 180 °𝑛 A × B = 0
- 79. Properties of Vector product 6 The vector product of a vector with itself is a null vector. A × A = AA sin 0 ° 𝑛 A × A = 0
- 80. Properties of Vector product 7 The vector product of two same unit vectors is a null vector. 𝑖 × 𝑖 = 𝑗 × 𝑗 = 𝑘 × 𝑘 = (1)(1) sin 0 °𝑛 = 0
- 81. Properties of Vector product 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
- 82. Aid to memory
- 83. Calculating vector product using components A = A𝑥𝑖 + A𝑦𝑗 + A𝑧𝑘 B = B𝑥𝑖 + B𝑦𝑗 + B𝑧𝑘 Let us have then A × B = A𝑥𝑖 + A𝑦𝑗 + A𝑧𝑘 × B𝑥𝑖 + B𝑦𝑗 + B𝑧𝑘 A × B = A𝑥𝑖 × B𝑥𝑖 + B𝑦𝑗 + B𝑧𝑘 + A𝑦𝑗 × B𝑥𝑖 + B𝑦𝑗 + B𝑧𝑘 + A𝑧𝑘 × B𝑥𝑖 + B𝑦𝑗 + B𝑧𝑘 = A𝑥B𝑥𝑖 × 𝑖 + A𝑥B𝑦𝑖 × 𝑗 + A𝑥B𝑧𝑖 × 𝑘 +A𝑦B𝑥𝑗 × 𝑖 + A𝑦B𝑦𝑗 × 𝑗 + A𝑦B𝑧𝑗 × 𝑘 +A𝑧B𝑥𝑘 × 𝑖 + A𝑧B𝑦𝑘 × 𝑗 + A𝑧B𝑧𝑘 × 𝑘 = A𝑥B𝑥(0) + A𝑥B𝑦(𝑘) + A𝑥B𝑧(−𝑗) +A𝑦B𝑥(−𝑘) + A𝑦B𝑦(0) + A𝑦B𝑧(𝑖) +A𝑧B𝑥(𝑗) + A𝑧B𝑦(−𝑖) + A𝑧B𝑧(0) = A𝑦B𝑧 𝑖 − A𝑧B𝑦 𝑖 + A𝑧B𝑥 𝑗 − A𝑥B𝑧(𝑗) + A𝑥B𝑦(𝑘) − A𝑦B𝑥(𝑘)
- 84. Calculating vector product using components A × B = 𝑖 A𝑦B𝑧 − A𝑧B𝑦 − 𝑗 A𝑥B𝑧 − A𝑧B𝑥 + 𝑘 (A𝑥B𝑦 − A𝑦B𝑥) A × B = 𝑖 𝑗 𝑘 A𝑥 A𝑦 A𝑧 B𝑥 B𝑦 B𝑧
- 85. Thank you