Cylindrical and spherical coordinates
Problem 9.375 · easy
Convert the rectangular point \( \displaystyle \left(0, - \frac{5 \sqrt{2}}{2}, - \frac{5 \sqrt{2}}{2}\right) \) to spherical coordinates (\( \displaystyle \rho > 0 \), \( \displaystyle 0 \le \theta < 2\pi \), \( \displaystyle 0 \le \varphi \le \pi \)).
- \[ 5 \]ρ = √(x² + y² + z²).✓ Proved
- \[ \frac{\left(-1\right) \sqrt{2}}{2} = - \frac{\sqrt{2}}{2} \]cos φ = z/ρ, so φ = 3*pi/4.✓ Proved
- \[ \left[\begin{matrix}0\\- \frac{5 \sqrt{2}}{2}\end{matrix}\right] \]θ = 3*pi/2 matches x and y.✓ Proved
Answer \( \left(5, \frac{3 \pi}{2}, \frac{3 \pi}{4}\right) \)
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 |
| 3 | ✓ 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: passqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly calculate the radius ρ, instead stating a tautology '5 = 5' without showing the derivation from the coordinates. Additionally, the justification for θ is presented as a matrix equality rather than a logical deduction from x and y.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to explicitly calculate the radius ρ, instead stating a tautology '5 = 5' without showing the derivation from the coordinates. Additionally, the justification for θ is presented as a matrix equality rather than a logical deduction from x and y.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to explicitly calculate or verify the radius \rho, \theta, and \phi from the given coordinates, instead presenting tautological equations that assume the answer is correct. It does not demonstrate the conversion process required by the problem statement.gpt-oss:20b: pass 2026-10-07
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 2026-10-07 with SymPy 1.14.0.