Cylindrical and spherical coordinates
Problem 9.218 · easy
Convert the rectangular point \( \displaystyle \left(-2, 2 \sqrt{3}, 0\right) \) to spherical coordinates (\( \displaystyle \rho > 0 \), \( \displaystyle 0 \le \theta < 2\pi \), \( \displaystyle 0 \le \varphi \le \pi \)).
- \[ 4 \]ρ = √(x² + y² + z²).✓ Proved
- \[ 0 \]cos φ = z/ρ, so φ = pi/2.✓ Proved
- \[ \left[\begin{matrix}-2\\2 \sqrt{3}\end{matrix}\right] \]θ = 2*pi/3 matches x and y.✓ Proved
Answer \( \left(4, \frac{2 \pi}{3}, \frac{\pi}{2}\right) \)
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 |
| 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: pass — The solution correctly identifies the spherical coordinates by verifying the radius, the polar angle (derived from z=0), and the azimuthal angle (derived from the quadrant of x and y). The algebraic checks are valid.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the spherical coordinates by verifying the radius, the polar angle (derived from z=0), and the azimuthal angle (derived from the quadrant of x and y). The algebraic checks are valid.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (misleading) 2026-10-04 — The solution provides no logical derivation, merely stating that the values match. It fails to explain how the quadrant of (-2, 2√3) determines θ = 2π/3, which is the critical conceptual step for students.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.