Planes: distances and angles
Problem 9.809 · medium
Find the angle between the planes \( \displaystyle - 3 x + 3 y - z = 2 \) and \( \displaystyle - 2 x - 3 y + 2 z = 5 \).
- The angle between planes is the (acute) angle between their normals.
- \[ \frac{\left|{3 \left(-3\right) - 1 \cdot 2 - 3 \left(-2\right)}\right|}{\sqrt{\left(-1\right)^{2} + \left(-3\right)^{2} + 3^{2}} \sqrt{\left(-2\right)^{2} + 2^{2} + \left(-3\right)^{2}}} = \frac{5 \sqrt{323}}{323} \]cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
- θ = arccos of that: acos(5*sqrt(323)/323).
Answer \( \operatorname{acos}{\left(\frac{5 \sqrt{323}}{323} \right)} \approx 73.85^\circ \)
Lines: 1 proved, 2 not checked. 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. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Not checked | — | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the normals' angle in floating point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The normal vectors extracted from the plane equations are incorrect. For -3x + 3y - z = 2, the normal is <-3, 3, -1>, but the solution uses <-1, -3, 3> (or similar permutation) in the norm calculation, leading to an incorrect denominator. Specifically, the norm calculation sqrt((-1)**2 + (-3)**2 + 3**2) corresponds to vector <-1, -3, 3>, which is not the normal of the first plane. The correct normal is <-3, 3, -1> with norm sqrt(19). Similarly, for -2x - 3y + 2z = 5, the normal is <-2, -3, 2> with norm sqrt(17), but the solution uses <-2, 2, -3> (norm sqrt(17) by coincidence of numbers, but wrong vector components for the dot product). The dot product calculation 3*(-3) - 1*2 - 3*(-2) also mixes components incorrectly.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The normal vectors extracted from the plane equations are incorrect. For -3x + 3y - z = 2, the normal is <-3, 3, -1>, but the solution uses <-1, -3, 3> (or similar permutation) in the norm calculation, leading to an incorrect denominator. Specifically, the norm calculation sqrt((-1)**2 + (-3)**2 + 3**2) corresponds to vector <-1, -3, 3>, which is not the normal of the first plane. The correct normal is <-3, 3, -1> with norm sqrt(19). Similarly, for -2x - 3y + 2z = 5, the normal is <-2, -3, 2> with norm sqrt(17), but the solution uses <-2, 2, -3> (norm sqrt(17) by coincidence of numbers, but wrong vector components for the dot product). The dot product calculation 3*(-3) - 1*2 - 3*(-2) also mixes components incorrectly.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly computes the dot product of the normal vectors. The normals are n1 = <-3, 3, -1> and n2 = <-2, -3, 2>. The dot product is (gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-18 — The normal vectors are incorrectly identified. The plane -3x + 3y - z = 2 has normal (-3, 3, -1), but the solution uses (-1, -3, 3) for the first norm and (-2, 2, -3) for the second, which corresponds to the plane -2x + 2y - 3z = 5, not -2x - 3y + 2z = 5. The second plane's normal should be (-2, -3, 2).gpt-oss:20b: pass 2026-10-18
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/planes_distance_angle, checked 2026-10-19 with SymPy 1.14.0.