Cylindrical and spherical coordinates
Problem 9.211 · easy
Convert the rectangular point \( \displaystyle \left(0, 5, -3\right) \) to cylindrical coordinates with \( \displaystyle r > 0 \), \( \displaystyle 0 \le \theta < 2\pi \).
- \[ 5 \]r = √(x² + y²).✓ Proved
- \[ \left[\begin{matrix}0\\5\end{matrix}\right] \]θ = pi/2 puts the point in the right quadrant.✓ Proved
Answer \( \left(5, \frac{\pi}{2}, -3\right) \)
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: fail (misleading) — The second sentence does not correctly justify why θ=π/2; it merely repeats the coordinates and claims the matrix equality, which does not explain the angle determination.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly calculate or justify the value of theta, instead presenting a tautological matrix equality that does not derive the angle from the coordinates. It also omits the trivial but necessary step of confirming z = -3.
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, instead presenting a tautological matrix equality that does not derive the angle from the coordinates. It also omits the trivial but necessary step of confirming z = -3.gpt-oss:20b: fail (misleading) 2026-10-04 — The second sentence does not correctly justify why θ=π/2; it merely repeats the coordinates and claims the matrix equality, which does not explain the angle determination.qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly calculate or state the value of r, relying on an identity '5=5' instead of showing r = sqrt(0^2 + 5^2) = 5. It also fails to explicitly calculate theta, merely stating that pi/2 is in the right quadrant without showing the arctan calculation or handling the x=0 case properly.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.