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Cylindrical and spherical coordinates

Problem 9.220 · easy

Convert the rectangular point \( \displaystyle \left(0, -6, 5\right) \) to cylindrical coordinates with \( \displaystyle r > 0 \), \( \displaystyle 0 \le \theta < 2\pi \).

\[ 6 \]
r = √(x² + y²).✓ Proved
\[ \left[\begin{matrix}0\\-6\end{matrix}\right] \]
θ = 3*pi/2 puts the point in the right quadrant.✓ Proved

Answer \( \left(6, \frac{3 \pi}{2}, 5\right) \)

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 or justify the value of \theta. It merely asserts that \theta = 3\pi/2 puts the point in the right quadrant without showing the calculation \tan^{-1}(y/x) or analyzing the signs of x and y to determine the correct angle in the specified range [0, 2\pi). While the final answer is correct, the reasoning provided is insufficient and skips the critical step of determining the angle.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly calculate or justify the value of \theta. It merely asserts that \theta = 3\pi/2 puts the point in the right quadrant without showing the calculation \tan^{-1}(y/x) or analyzing the signs of x and y to determine the correct angle in the specified range [0, 2\pi). While the final answer is correct, the reasoning provided is insufficient and skips the critical step of determining the angle.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly calculate or state the conversion for the z-coordinate, which is required for cylindrical coordinates. Additionally, the second line is a nonsensical matrix equality that does not constitute a valid derivation for theta.
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.