Lecture Notes: Vectors and Motion in a Plane

Introduction to Vectors

• Introduction to Vectors: Any quantity defined by magnitude and direction.
• Definition of Vector: Quantity with both magnitude and direction.
• Vector Algebra:
  • Addition of Vectors
  • Subtraction of Vectors
  • Multiplication of Vectors
  • Division of Vectors

Classification of Vector

• Scalar: Only magnitude, no direction
• Vector: Magnitude + direction

Representation of Vectors

• Representation: A vector is represented by an arrow.
• Components:
  • In terms of i-cap, j-cap, k-cap
  • In x, y, z coordinates

Mathematical Representation of Vectors

• Mathematical Representation:
  • In terms of i, j, k
  • 3i + 4j + 5k

Unit Vector

• Definition: Indicates direction, magnitude 1
• Calculation: Divide the vector by its magnitude

Vector Multiplication

• Dot Product: a.b = |a||b|cos(theta)
• Cross Product: a×b = |a||b|sin(theta)n

Motion in a Plane

• Motion in a Plane: Motion in two dimensions
• Projectile Motion:
  • Hmax (maximum height)
  • Time of Flight
  • Range
• Uniform Circular Motion:
  • Constant speed, changing direction

Equations of Vectors and Motion

• Position Vector: r(t) = x(t)i + y(t)j
• Velocity: v = dr/dt
• Acceleration: a = dv/dt
• Vector Equation: v = u + at

Uniform Circular Motion

• Definition: Motion along a circular path
• Centripetal Force: Force pulling towards the center
• Centripetal Acceleration: v²/r

Angular Quantities

• Angular Displacement: θ
• Angular Velocity: ω = dθ/dt
• Angular Acceleration: α = dω/dt

Key Points

• Angular Velocity: ω = v/r
• Tangential Acceleration: a_t = rα
• Centripetal Acceleration: a_c = ω²r

Summary

• Motion Equations:
  • Linear: v = v₀ + at, s = v₀t + ½at²
  • Angular: ω = ω₀ + αt, θ = ω₀t + ½αt²
• Position, Velocity, Acceleration
• Uniform Motion: Constant ω
• Centripetal Motion: a_c = v²/r or a_c = ω²r