Resultant from Event Samples — Homework (Ideas Only)

Given (per item): a list of angles (degrees) observed in one time window for a single system.

Task: compute the resultant vector and derived quantities, with and to isolate ideas.

Pipeline

Notes. Evaluate trig in radians; display angles in degrees. Only the sine projection reaches the denominator; the cosine projection never does. Here is radius and is Surprise.

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Report for each item

and an “alignment” score with the rule of thumb: Strong , Moderate , Weak .

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Problems (angles in degrees)

1. MLK-like cluster
   {68, 72, 75, 77, 70, 73, 74, 69, 76, 71}
2. Everyday circles
   {25, 28, 30, 32, 29, 31, 27, 33, 26, 34}
3. Perfect-storm orientation
   {90, 90, 90, 90, 90, 90, 90, 90, 90, 90}
4. Pure realization
   {0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
5. Balanced opposite crowds
   {0, 0, 0, 0, 0, 180, 180, 180, 180, 180}
6. Uniform decagon (full spread)
   {0, 36, 72, 108, 144, 180, 216, 252, 288, 324}
7. Small angle, strong
   {10, 8, 12, 9, 11, 7, 13, 10, 8, 12}
8. Split at ±30° (equal counts)
   {30, 30, 30, 30, 30, −30, −30, −30, −30, −30}
9. Lopsided 30° vs 210°
   {30, 30, 30, 30, 30, 30, 210, 210, 210, 210}
10. Three-fold symmetry
    {0, 0, 0, 0, 120, 120, 120, 120, 240, 240, 240, 240}

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Answer Key (rounded to three decimals)

1. N = 10,  C = 9.987,  φ = 72.500°,  j = 3.003,  k = 9.525,  |E| = 9.578,  r = 0.104,  S = −2.259,  alignment = 0.954 → Strong.
   Imaginary component in denominator: k = 9.525.
2. N = 10,  C = 9.987,  φ = 29.500°,  j = 8.693,  k = 4.918,  |E| = 5.019,  r = 0.199,  S = −1.613,  alignment = 0.492 → Moderate.
   Imaginary component: k = 4.918.
3. N = 10,  C = 10.000,  φ = 90.000°,  j = 0.000,  k = 10.000,  |E| = 10.050,  r = 0.100,  S = −2.308,  alignment = 1.000 → Strong.
   Imaginary component: k = 10.000.
4. N = 10,  C = 10.000,  φ = 0.000°,  j = 10.000,  k = 0.000,  |E| = 1.000,  r = 1.000,  S = 0.000,  alignment = 0.000 → Weak.
   Imaginary component: k = 0.000.
5. N = 10,  C = 0.000,  φ = undefined,  j = 0.000,  k = 0.000,  |E| = 1.000,  r = 1.000,  S = 0.000,  alignment = 0.000 → Weak.
   Perfect cancellation in this lens.
6. N = 10,  C = 0.000,  φ = undefined,  j = 0.000,  k = 0.000,  |E| = 1.000,  r = 1.000,  S = 0.000,  alignment ≈ 0.000 → Weak.
   Full rotational symmetry cancels.
7. N = 10,  C = 9.995,  φ = 10.000°,  j = 9.843,  k = 1.736,  |E| = 2.003,  r = 0.499,  S = −0.695,  alignment = 0.174 → Weak.
   Imaginary component: k = 1.736.
8. N = 10,  C = 8.660,  φ = 0.000°,  j = 8.660,  k = 0.000,  |E| = 1.000,  r = 1.000,  S = 0.000,  alignment ≈ 0.000 → Weak.
   Vertical parts cancel; horizontal parts add along 0°.
9. N = 10,  C = 2.000,  φ = 30.000°,  j = 1.732,  k = 1.000,  |E| = 1.414,  r = 0.707,  S = −0.347,  alignment = 0.500 → Moderate.
   Imaginary component: k = 1.000.
10. N = 12,  C = 0.000,  φ = undefined,  j = 0.000,  k = 0.000,  |E| = 1.000,  r = 1.000,  S = 0.000,  alignment = 0.000.
    Three equal clusters at 0°, 120°, 240° cancel exactly (classification not meaningful when C = 0).

Reminder. Add unit arrows first; split once; only reaches the denominator. If report and note . To detect a two-fold split, recompute with doubled angles before summing; for three-fold, triple the angles.