Lecture Notes: Vectors and Directional Angles

Magnitude of V

• Magnitude Calculation:
  • Formula: |V| = (\sqrt{V_1^2 + V_2^2})
  • Given V in linear combination: V = V1(i) + V2(j)
  • Example:
    • V1 = 6, V2 = -6
    • Calculation: (\sqrt{6^2 + (-6)^2}) = (\sqrt{72})
• Simplifying (\sqrt{72}):
  • Find the largest square number that divides 72.
  • 72 = 36 * 2
  • Thus, (\sqrt{72} = 6\sqrt{2})
  • Breakdown:
    • Alternative methods: 72 = 12 * 6 and further break down as needed.

Directional Angle

• **Calculating Directional Angle: **
  • Directional angle (\theta) using tangent:
    • (\tan(\theta) = \frac{V_2}{V_1} = \frac{-6}{6} = -1)
    • (\theta = \tan^{-1}(-1))
  • Result: (\theta = -45°)
  • Converting:
    • Negative angle (-45°) to positive direction by adding 360°: (315°)
    • Thus, (\theta = 315°)

Summary

• Magnitude of vector V: (6\sqrt{2})
• Directional angle (\theta): 315°