Lecture on Adding Vectors

Key Concepts

• Vector: A quantity with both magnitude and direction.
• Magnitude: The size or length of the vector.
• Direction: The orientation of the vector in space.

Adding Parallel Vectors

• Same Direction: Sum magnitudes directly.
  • Example: 100 N east + 50 N east = 150 N east.
• Opposite Directions: Subtract magnitudes, direction remains of the larger magnitude.
  • Example: 200 N east - 120 N west = 80 N east.
  • Example: 60 N east - 90 N west = -30 N or 30 N west.

Adding Perpendicular Vectors

• Using Pythagoras' Theorem: Resultant vector is the hypotenuse.
  • Example: 30 N east + 40 N north results in √(30² + 40²) = 50 N.
  • Direction: Use inverse tangent (tan⁻¹) to find angle.
    • θ = tan⁻¹(opposite/adjacent).
    • Example: θ = tan⁻¹(40/30) = 53.1 degrees relative to x-axis.

Examples of Perpendicular Vectors

• West and South: 50 N west + 120 N south.
  • Magnitude: √(50² + 120²) = 130 N.
  • Direction: θ = tan⁻¹(120/50) -> Reference Angle = 67.4 degrees.
    • Total Angle = 180 + 67.4 = 247.4 degrees.
• East and South: 45 N east + 60 N south.
  • Magnitude: √(45² + 60²) = 75 N.
  • Direction: θ = tan⁻¹(60/45) -> Reference Angle = 53.1 degrees.
    • Total Angle = 360 - 53.1 = 306.9 degrees.

Angles and Quadrants

• Quadrant Determination: Reference angle determined by inverse tangent.
  • Quadrant 1: Angle = Reference Angle.
  • Quadrant 2: Angle = 180 - Reference Angle.
  • Quadrant 3: Angle = 180 + Reference Angle.
  • Quadrant 4: Angle = 360 - Reference Angle.

Adding Non-Parallel or Non-Perpendicular Vectors

• Resolve into Components:
  • Example: 100 N east + 150 N at 30 degrees above x-axis.
    • Component Formulas:
      • Fx = Fcos(θ)
      • Fy = Fsin(θ)
    • For 100 N east (0°):
      • Fx = 100 cos(0) = 100 N
      • Fy = 100 sin(0) = 0 N
    • For 150 N at 30°:
      • Fx = 150 cos(30) = 129.9 N
      • Fy = 150 sin(30) = 75 N
    • Sum of Components:
      • Fx_total = 100 + 129.9 = 229.9 N
      • Fy_total = 0 + 75 = 75 N
• Resultant Vector:
  • Magnitude using Pythagoras: √(Fx_total² + Fy_total²) = 241.8 N
  • Angle using inverse tangent: θ = tan⁻¹(Fy_total / Fx_total) = 18.1°

Summary

• Adding Vectors: Consider direction (parallel, anti-parallel, or perpendicular), resolve into components if necessary, and use Pythagorean theorem and trigonometric functions to find the resultant vector's magnitude and direction.