Physics Vectors and Arithmetic Overview

Jul 11, 2024

Lecture Notes: Physics Vectors and Arithmetic Topics

Introduction

  • Physical Quantity: Two types - Scalar and Vector
  • Scalar - Only magnitude (example - mass, temperature)
  • Vector - Magnitude and direction (example - displacement, velocity)

Representation of Vector

  • Symbolic Form: Arrow above the vector
  • Graphical Representation: Arrowhead
  • Unit Vector: Magnitude of 1

Vector Angles

  • Tail to Tail and Head to Head both are valid
  • Always take the smaller angle

Types of Vectors

  1. Equal Vectors: Same magnitude and direction
  2. Parallel Vectors: Same magnitude and direction
  3. Anti-parallel Vectors: Opposite direction, θ = 180°
  4. Unit Vector: Magnitude = 1
  5. Co-initial Vectors: Start from the same point
  6. Co-linear Vectors: Lie on the same line, θ = 0° or 180°
  7. Coplanar Vectors: Lie on the same plane
  8. Polar Vector: Based on initial and final position
  9. Axial Vector: Represents rotational effect

Shifting and Rotation of Vector

  • Shifting: Vector's magnitude and direction should not change
  • Rotation: If rotated 360°, the vector remains the same

Resolution of Vector

  • Vector components:
    • x Component: A cosθ
    • y Component: A sinθ
    • z Component: A cosγ

Vector Operations

  • Dot Product: A . B = AB cosθ
  • Cross Product: A × B = AB sinθ n^ (new vector formed)
  • Projection of Vector: A . B/|B|
  • Important Points of Dot Product:
    • A . A = |A|^2
    • A . (-B) = -(A . B)
    • A . B = 0 when vectors are perpendicular

Triangle Law of Addition

  • Methods of adding vectors according to version:
    • Triangle Law: Only for two vectors
    • Parallelogram Law: Only for two vectors
    • Polygon Law: For more than two vectors

Power of Envelopes

  • Addition:
    • Commutative: A + B = B + A
    • Associative: A + (B + C) = (A + B) + C
  • Subtraction: R = A - B
  • Range of Resultant: |A - B| ≤ R ≤ |A + B|

Quadratic Formulas

  • Generalized form: ax^2 + bx + c = 0
  • Solution formula: x = [-b ± sqrt(b^2 - 4ac)] / 2a

Concepts of Logs and Nominals

  • Log Format: log_b(x)
  • Natural Log: Base e (ln e)
  • Common Log: Base 10 (log 10)
  • Important Formulas:
    • log_b(m^n) = n log_b(m)
    • log_b(xy) = log_b(x) + log_b(y)
    • log_b(x/y) = log_b(x) - log_b(y)