Cylindrical and spherical coordinates
Problem 9.380 · easy
Convert the rectangular point \( \displaystyle \left(\frac{3 \sqrt{2}}{2}, - \frac{3 \sqrt{2}}{2}, 0\right) \) to cylindrical coordinates with \( \displaystyle r > 0 \), \( \displaystyle 0 \le \theta < 2\pi \).
- \[ 3 \]r = √(x² + y²).✓ Proved
- \[ \left[\begin{matrix}\frac{3 \sqrt{2}}{2}\\- \frac{3 \sqrt{2}}{2}\end{matrix}\right] \]θ = 7*pi/4 puts the point in the right quadrant.✓ Proved
Answer \( \left(3, \frac{7 \pi}{4}, 0\right) \)
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: passqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly calculate or justify the value of theta, merely stating that 7pi/4 puts the point in the right quadrant without showing the arctan calculation or quadrant analysis. It also omits the z-coordinate conversion, although it is trivial.
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 or justify the value of theta, merely stating that 7pi/4 puts the point in the right quadrant without showing the arctan calculation or quadrant analysis. It also omits the z-coordinate conversion, although it is trivial.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to explicitly calculate or justify the angle theta, merely stating the result without showing the arctan calculation or quadrant analysis. It also omits the z-coordinate conversion, which is trivial but required for a complete cylindrical coordinate tuple.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.