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Cylindrical and spherical coordinates

Problem 9.217 · easy

Convert the rectangular point $ \displaystyle \left(- \frac{5 \sqrt{2}}{2}, \frac{5 \sqrt{2}}{2}, -4\right) $ to cylindrical coordinates with $ \displaystyle r > 0 $, $ \displaystyle 0 \le \theta < 2\pi $.

\[ 5 \]
r = √(x² + y²).✓ Proved
\[ \left[\begin{matrix}- \frac{5 \sqrt{2}}{2}\\\frac{5 \sqrt{2}}{2}\end{matrix}\right] \]
θ = 3*pi/4 puts the point in the right quadrant.✓ Proved

Answer $ \left(5, \frac{3 \pi}{4}, -4\right) $

Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	converted back to rectangular in floating point

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly calculate the radius r or the angle theta, instead presenting tautologies (5=5 and vector=vector). It does not demonstrate that the conversion formulas were applied correctly to the given coordinates.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly calculate the radius r or the angle theta, instead presenting tautologies (5=5 and vector=vector). It does not demonstrate that the conversion formulas were applied correctly to the given coordinates.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution fails to explicitly calculate the radius $r$ and angle $\theta$ from the given coordinates, instead presenting tautological equations.
gpt-oss:20b: pass 2026-10-04

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/cylindrical_spherical_coords, checked 2026-10-04 with SymPy 1.14.0.