Planes: distances and angles
Problem 9.881 · medium
Find the angle between the planes \( \displaystyle - 2 x - y + 2 z = -4 \) and \( \displaystyle - 3 x + 3 y - 2 z = 1 \).
- The angle between planes is the (acute) angle between their normals.
- \[ \frac{\left|{2 \left(-2\right) - 1 \cdot 3 - 2 \left(-3\right)}\right|}{\sqrt{\left(-1\right)^{2} + \left(-2\right)^{2} + 2^{2}} \sqrt{\left(-2\right)^{2} + \left(-3\right)^{2} + 3^{2}}} = \frac{\sqrt{22}}{66} \]cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
- θ = arccos of that: acos(sqrt(22)/66).
Answer \( \operatorname{acos}{\left(\frac{\sqrt{22}}{66} \right)} \approx 85.92^\circ \)
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 -2x - y + 2z = -4, the normal is <-2, -1, 2>, but the solution uses <-1, -2, 2> in the denominator calculation. Similarly, for -3x + 3y - 2z = 1, the normal is <-3, 3, -2>, but the solution uses <-2, -3, 3>. This leads to an incorrect cosine value and angle.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-11-05 — The normal vectors extracted from the plane equations are incorrect. For -2x - y + 2z = -4, the normal is <-2, -1, 2>, but the solution uses <-1, -2, 2> in the denominator calculation. Similarly, for -3x + 3y - 2z = 1, the normal is <-3, 3, -2>, but the solution uses <-2, -3, 3>. This leads to an incorrect cosine value and angle.gpt-oss:20b: pass 2026-11-05qwen3.6:27b-mlx: fail (error) 2026-11-04 — The normal vectors extracted from the plane equations are incorrect. For -2x - y + 2z = -4, the normal is <-2, -1, 2>, but the denominator calculation uses components <-1, -2, 2>. Similarly, for -3x + 3y - 2z = 1, the normal is <-3, 3, -2>, but the denominator uses <-2, -3, 3>. The dot product in the numerator also uses incorrect components (2, 1, 2) instead of (-2, -1, 2). While the final numerical answer happens to be correct due to symmetry in magnitudes and absolute values, the intermediate steps demonstrate a fundamental misunderstanding of how to extract normal vectors from plane equations.gpt-oss:20b: pass 2026-11-04qwen3.6:27b-mlx: fail (error) 2026-11-04 — The normal vectors extracted from the plane equations are incorrect. For -2x - y + 2z = -4, the normal is <-2, -1, 2>, but the denominator uses components <-1, -2, 2>. For -3x + 3y - 2z = 1, the normal is <-3, 3, -2>, but the denominator uses <-2, -3, 3>. The dot product numerator also uses inconsistent components.gpt-oss:20b: pass 2026-11-04
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/planes_distance_angle, checked 2026-11-05 with SymPy 1.14.0.