Efficient Calculation of Final Magnitude and Angle in Vector Addition Method

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SUMMARY

The discussion centers on a method for efficiently calculating the final magnitude (Lₜ) and angle (θₜ) in vector addition, particularly relevant for electricians. The proposed formula, ∑(cos(θₙ-θₜ)⋅Lₙ)=Lₜ, simplifies the calculation of Lₜ using Euclid's axiom of angles within parallel lines. The conversation highlights the challenge of deriving θₜ from this formula and explores alternative methods, including the use of Pythagorean theorem and complex number representations. The participants emphasize the need for clarity and derivation steps to validate the proposed shortcuts.

PREREQUISITES
  • Understanding of vector addition and trigonometry
  • Familiarity with complex number representation in electrical engineering
  • Knowledge of Pythagorean theorem in the context of vector magnitudes
  • Basic skills in using mathematical notation, including LaTeX
NEXT STEPS
  • Research "Complex number representation in electrical circuits" for enhanced understanding of vector calculations
  • Study "Pythagorean theorem applications in vector addition" to solidify foundational concepts
  • Learn "Using Excel for vector calculations" to streamline the computation process in practical scenarios
  • Explore "Deriving angles in vector addition" to improve methods for calculating θₜ
USEFUL FOR

This discussion is beneficial for electricians, electrical engineers, and students in physics or engineering fields who are involved in vector calculations and seeking efficient methods for determining magnitudes and angles in electrical systems.

  • #31
okay this is what I can get one my own

$$ \left( \sum (L_n cos(\theta_n - \theta_t)) \right)^2 = \left( \sum (L_n sin (\theta_n)) \right)^2 + \left( \sum (L_n cos(\theta_n)) \right)^2 $$

$$ \left( \sum (L_n cos(\theta_n - \theta_t)) \right)^2 / \left( \sum (L_n) \right)^2 = \left( \left( \sum (L_n sin (\theta_n)) \right)^2 + \left( \sum (L_n cos(\theta_n)) \right)^2 \right) / \left( \sum (L_n) \right)^2 $$

$$ \left( \sum (cos(\theta_n) cos(\theta_t) + sin(\theta_n) sin(\theta_t)) \right)^2 = \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 $$

$$ \left( \sum (cos(\theta_n) cos(\theta_t)) + \sum (sin(\theta_n) sin(\theta_t)) \right)^2 = \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 $$

$$ \left( cos(\theta_t) \sum (cos(\theta_n)) + sin(\theta_t) \sum (sin(\theta_n)) \right)^2 = \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 $$

$$ \left( cos(\theta_t) \sum (cos(\theta_n)) \right)^2 +2 cos(\theta_t) \sum (cos(\theta_n)) sin(\theta_t) \sum (sin(\theta_n)) + \left( sin(\theta_t) \sum (sin(\theta_n)) \right)^2 = \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 $$
 
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  • #32
$$ \left( \left( cos(\theta_t) \sum (cos(\theta_n)) \right)^2 +2 cos(\theta_t) \sum (cos(\theta_n)) sin(\theta_t) \sum (sin(\theta_n)) + \left( sin(\theta_t) \sum (sin(\theta_n)) \right)^2 \right) / \left( \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 \right) = \left( \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 \right) / \left( \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 \right) $$

$$ \left( \left( cos(\theta_t) \sum (cos(\theta_n)) \right)^2 \right) / \left( \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 \right) + \left( 2 cos(\theta_t) \sum (cos(\theta_n)) sin(\theta_t) \sum (sin(\theta_n)) \right) / \left( \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 \right) + \left( \left( sin(\theta_t) \sum (sin(\theta_n)) \right)^2 \right) / \left( \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 \right) = 1 $$

$$ \left( cos(\theta_t) \right)^2 / \left( \left( \sum (sin (\theta_n)) \right)^2 + 1 \right) + \left( 2 cos(\theta_t) sin(\theta_t) \right) / \left( \left( \sum (sin (\theta_n)) \right) + \left( \sum (cos(\theta_n)) \right) \right) + \left( sin(\theta_t) \right)^2 / \left( 1 + \left( \sum (cos(\theta_n)) \right)^2 \right) = 1 $$
 
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  • #33
$$ \left( \left( cos(\theta_t) \sum (cos(\theta_n)) \right)^2 + 2 cos(\theta_t) \sum (cos(\theta_n)) sin(\theta_t) \sum (sin(\theta_n)) + \left( sin(\theta_t) \sum (sin(\theta_n)) \right)^2 \right) / \left( \sum (sin (\theta_n)) \sum (cos(\theta_n)) \right) = \left( \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 \right) / \left( \sum (sin (\theta_n)) \sum (cos(\theta_n)) \right) $$

$$ \left( \left( cos(\theta_t) \sum (cos(\theta_n)) \right)^2 \right) / \left( \sum (sin (\theta_n)) \sum (cos(\theta_n)) \right) + \left( 2 cos(\theta_t) \sum (cos(\theta_n)) sin(\theta_t) \sum (sin(\theta_n)) \right) / \left( \sum (sin (\theta_n)) \sum (cos(\theta_n)) \right) + \left( \left( sin(\theta_t) \sum (sin(\theta_n)) \right)^2 \right) / \left( \sum (sin (\theta_n)) \sum (cos(\theta_n)) \right) = \left( \sum (sin (\theta_n)) \right)^2 / \left( \sum (sin (\theta_n)) \sum (cos(\theta_n)) \right) + \left( \sum (cos(\theta_n)) \right)^2 / \left( \sum (sin (\theta_n)) \sum (cos(\theta_n)) \right) $$

$$ \left( cos(\theta_t) \right)^2 \sum (cot(\theta_n)) + 2 cos(\theta_t) sin(\theta_t) + \left( sin(\theta_t) \right)^2 \sum (tan(\theta_n)) = \sum (tan (\theta_n)) + \sum (cot(\theta_n)) $$
 
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  • #34
found an error in my equation, this is fixed

$$ \frac {\left( cos(\theta_t) \sum (L_n cos(\theta_n)) \right)^2 + 2 cos(\theta_t) \sum (L_n cos(\theta_n)) sin(\theta_t) \sum (L_n sin(\theta_n)) + \left( sin(\theta_t) \sum (L_n sin(\theta_n)) \right)^2} {\sum (L_n sin (\theta_n)) \sum (L_n cos(\theta_n))} = \frac {\left( \sum (L_n sin (\theta_n)) \right)^2 + \left( \sum (L_n cos(\theta_n)) \right)^2} {\sum (L_n sin (\theta_n)) \sum (L_n cos(\theta_n))} $$

$$ \frac {\left( cos(\theta_t) \sum (L_n cos(\theta_n)) \right)^2} {\sum (L_n sin (\theta_n)) \sum (L_n cos(\theta_n))} + \frac {2 cos(\theta_t) \sum (L_n cos(\theta_n)) sin(\theta_t) \sum (L_n sin(\theta_n))} {\sum (L_n sin (\theta_n)) \sum (L_n cos(\theta_n))} + \frac {\left( sin(\theta_t) \sum (L_n sin(\theta_n)) \right)^2} {\sum (L_n sin (\theta_n)) \sum (L_n cos(\theta_n))} = \frac {\left( \sum (L_n sin (\theta_n)) \right)^2} {\sum (L_n sin (\theta_n)) \sum (L_n cos(\theta_n))} + \frac {\left( \sum (L_n cos(\theta_n)) \right)^2} {\sum (L_n sin (\theta_n)) \sum (L_n cos(\theta_n))} $$

$$ cos ^2 (\theta_t) \frac {\sum L_n cos(\theta_n)} {\sum L_n sin (\theta_n)} + 2 cos(\theta_t) sin(\theta_t) + sin ^2 (\theta_t) \frac {\sum L_n sin(\theta_n)} {\sum L_n cos(\theta_n)} = \frac {\sum L_n sin (\theta_n)} {\sum L_n cos(\theta_n)} + \frac {\sum L_n cos(\theta_n)} {\sum L_n sin (\theta_n)} $$
 
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