Physics Chapter 3: Vector Analysis

Course Content

• Course covers Physics, with chapters including Physical Quantity, Units, and Dimensions (1 week); Motion in One Dimension (2 weeks); Vector Analysis (2 weeks); Waves, Oscillations, and Sound (2 weeks); Light, Lenses, and Mirrors (2 weeks); Heat and Thermodynamics (2 weeks); and Electricity & Magnetism (2 weeks).

Chapter 3: Vector Analysis

• Vectors & Physics: Vectors have magnitude and direction. A vector is represented as F = 7i − 5j + 3k. Components of a vector are calculated using trigonometry (Fx = F cos θ; Fy = F sin θ). Vectors are used to find the resultant vectors.
• Coordinate Systems:
  • Cartesian (rectangular): Points are labeled (x, y).
  • Plane polar: Points are labeled (r, θ).
• Relations Between Systems: Conversions between Cartesian and polar coordinates:
  • x = r cos θ
  • y = r sin θ
  • r = √(x² + y²)
  • θ = tan⁻¹(y/x)
• Vector and Scalar Quantities:
  • Scalar: Only magnitude (e.g., temperature).
  • Vector: Magnitude and direction (e.g., velocity).
• Vectors' Notations: Text-book notations (bold) and other notations (with arrows).
• Displacement Vector: The magnitude of the displacement vector is the shortest distance between the end points (A and B), regardless of the trajectory.
• Vector Properties:
  • Equality: Two vectors are equal if their magnitudes are equal and they point in the same direction.
  • Addition (Triangle Method, Parallelogram Method): methods for adding vectors.
  • Addition: (More than two vectors) Polygon Method.
  • Commutative Law: A + B = B + A
  • Associative Law: (A + B) + C = A + (B + C)
  • Negative of a vector: The negative of a vector has the same magnitude but points in the opposite direction.
• Subtraction: A - B = A + (-B)
• Multiplication: The scalar multiple of a vector (m * A) is parallel or antiparallel to A with magnitude |m| * |A|.
• Components of Vectors and Unit Vectors:
  • A vector's components are its projections along coordinate axes (e.g., ax, ay).
  • Unit vector notations are for directions (î, ĵ, k).
  • Vector components are Ax=A cos θ; Ay = A sin θ; Direction of A = tan⁻¹ (Ay/Ax)
• Sum of Vectors Using Components: Vectors are added by adding their corresponding components. R = Ax + Bx, Ry = Ay + By. 
• Vectors in Three Dimensions: Vectors are expressed as A = Ax î + Ay ĵ + Az k. Vectors are added component-wise.

Additional Topics

• Application Examples: Work problems involving displacement vector calculations (e.g., car travel). Given Cartesian coordinates to find magnitude and direction. Problem examples from the homework exercises.
• Quick Quizzes: Identify scalar vs. vector quantities, possible magnitudes of sums, and conditions for resultant vector magnitude.
• Homework Problems: Specific problems to solve (e.g., HW-3 Serway Version 8 Problems 5, 15, 31, 37 and 43).

This quiz covers Chapter 3 on Vector Analysis in Physics. It includes the fundamentals of vectors, the relationship between different coordinate systems, and the conversion formulas used in physics. Test your understanding of vector and scalar quantities.