Cylindrical and spherical coordinates
Problem 9.373 · easy
Convert the cylindrical point \( \displaystyle (r, \theta, z) = \left(3, \frac{\pi}{2}, -2\right) \) to rectangular coordinates.
- \[ \left[\begin{matrix}0\\3\\-2\end{matrix}\right] \]x = r cos θ, y = r sin θ, z = z.✓ Proved
Answer \( \left(0, 3, -2\right) \)
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 |
| 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: pass — The solution correctly applies the conversion formulas x = r cos(θ), y = r sin(θ), z = z to obtain the correct rectangular coordinates.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies the conversion formulas x = r cos(θ), y = r sin(θ), z = z to obtain the correct rectangular coordinates.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution provides no derivation or explanation, merely stating the final result as an identity. It fails to show the application of the conversion formulas x = r cos(θ) and y = r sin(θ) with the given values.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.