Lecture Notes on Vectors

Jul 29, 2024

Lecture Notes on Vectors

Introduction

  • Welcome by Abhishek Sahu
  • Topic: Vectors in a short concise manner.
  • Aim: Clear confusion about vectors among students.
  • Request for feedback/comments for future videos.

Physical Quantity

  • Definition: Measurable quantities.
    • Examples: Mass, Length, Time, Temperature, Power, Force, Velocity.
  • Categories:
    • Scalar Quantities (Magnitude only, no direction)
    • Vector Quantities (Both magnitude and direction)

Scalar vs Vector Quantities

  • Scalar Quantities: Have magnitude but no direction (e.g., mass, temperature, speed).
  • Vector Quantities: Have both magnitude and direction (e.g., force, velocity).

Magnitude

  • Magnitude: Represents the amount (Example: 10 kg potatoes).
  • Importance of direction in vectors affects the resultant.

Addition of Vectors

  • Scalar Addition: Normal adding (e.g. mass or time).
  • Vector Addition: Different methods to add vectors depending on direction.

Methods of Vector Addition

  1. Algebraic Addition: Adding in the same direction or subtracting in opposite directions.
  2. Triangle Law: Connect tail of one vector to head of another.
  3. Parallelogram Law: Use tails of both vectors to form a parallelogram.
  4. Polygon Law: For multiple vectors, add them sequentially in a polygonal form.

Resultant of Vectors

  • Resultant can range between minimum (0) and maximum (sum of magnitudes).
  • Example: Resultant of two 10 Newton forces at different angles calculated using:
    • Formula: R = √(A² + B² + 2ABcosθ).

Types of Vectors

  • Equal Vectors: Same magnitude and direction.
  • Negative Vectors: Same magnitude but opposite direction.
  • Position Vectors: Describe position with respect to origin.
  • Displacement Vectors: Change in position from one point to another.
  • Free Vectors: Can shift without changing magnitude or direction.
  • Unit Vectors: Have a magnitude of one and indicate direction (e.g., î, ĵ, k̂).

Multiplication of Vectors

  • Dot Product (Scalar Product): Vector multiplication resulting in a scalar.
  • Example: Work = Force × Displacement (Scalar Quantity).
  • Cross Product (Vector Product): Vector multiplication resulting in another vector.

Scalar Product Formula

  • A·B = |A||B|cos(θ)
  • Example for unit vectors: i·i = 1, j·j = 1, and others = 0.

Cross Product Formula

  • A x B = |A||B|sin(θ)
  • Positive results for counterclockwise rotation and negative for clockwise.

Resolution of Vectors

  • Breakdown of a vector into components, usually perpendicular to each other (e.g. horizontal and vertical).
  • Components are found using trigonometric functions (sin, cos).
  • For example, if vector has a magnitude of 10N at an angle of 30 degrees:
    • Horizontal component = 10cos(30)
    • Vertical component = 10sin(30)

Homework Assignment

  • Find unit vector of the vector product A x B.
  • Method: Unit vector formula = A x B / |A x B|.

Notes

  • Students encouraged to share video and join Telegram group for updates.