Cylindrical and spherical coordinates
Problem 9.533 · easy
Convert the rectangular point \( \displaystyle \left(0, - \frac{\sqrt{3}}{2}, \frac{1}{2}\right) \) to spherical coordinates (\( \displaystyle \rho > 0 \), \( \displaystyle 0 \le \theta < 2\pi \), \( \displaystyle 0 \le \varphi \le \pi \)).
- \[ 1 \]ρ = √(x² + y² + z²).✓ Proved
- \[ 1 \cdot \frac{1}{2} = \frac{1}{2} \]cos φ = z/ρ, so φ = pi/3.✓ Proved
- \[ \left[\begin{matrix}0\\\frac{\left(-1\right) \sqrt{3}}{2}\end{matrix}\right] = \left[\begin{matrix}0\\- \frac{\sqrt{3}}{2}\end{matrix}\right] \]θ = 3*pi/2 matches x and y.✓ Proved
Answer \( \left(1, \frac{3 \pi}{2}, \frac{\pi}{3}\right) \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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 conversion formulas for rho, phi, and theta against the given rectangular point. The logic is sound and the final answer is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly identifies the spherical coordinates by verifying the conversion formulas for rho, phi, and theta against the given rectangular point. The logic is sound and the final answer is correct.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly identifies the spherical coordinates. The equations verify the magnitude, the polar angle (via z-component), and the azimuthal angle (via x,y components) consistently with the given point and standard conventions.gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.