Efficient Calculation of Final Magnitude and Angle in Vector Addition Method

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The discussion centers on a method for efficiently calculating the final magnitude (Lₜ) and angle (θₜ) in vector addition, primarily using Euclid's axiom related to angles in parallel lines. The proposed formula for Lₜ is based on the cosine of the angle differences, but the challenge remains in determining θₜ without prior knowledge of it. Participants express confusion over the derivation steps and the need for clarity in the calculations, suggesting that traditional methods like complex numbers or Cartesian coordinates might be more straightforward. There is a consensus that while shortcuts are desirable, vector addition typically requires term-by-term calculations unless specific relationships exist. The conversation highlights the need for a clearer presentation of the formulas and their applications in practical scenarios, especially in electrical engineering contexts.
  • #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 equasion, 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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