Planes: distances and angles
Problem 9.811 · medium
Find the angle between the planes \( \displaystyle - x + 4 y - 4 z = -2 \) and \( \displaystyle - 3 x - 3 y - 2 z = -2 \).
- The angle between planes is the (acute) angle between their normals.
- \[ \frac{\left|{4 \left(-3\right) - 1 \left(-3\right) - 4 \left(-2\right)}\right|}{\sqrt{\left(-1\right)^{2} + 4^{2} + \left(-4\right)^{2}} \sqrt{\left(-2\right)^{2} + \left(-3\right)^{2} + \left(-3\right)^{2}}} = \frac{\sqrt{6}}{66} \]cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
- θ = arccos of that: acos(sqrt(6)/66).
Answer \( \operatorname{acos}{\left(\frac{\sqrt{6}}{66} \right)} \approx 87.87^\circ \)
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 vector for the second plane is incorrectly identified as (-2, -3, -3) instead of (-3, -3, -2), leading to an incorrect dot product and magnitude calculation.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The normal vector for the second plane is incorrectly identified as (-2, -3, -3) instead of (-3, -3, -2), leading to an incorrect dot product and magnitude calculation.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-18 — The normal vector for the second plane is incorrectly identified as (-2, -3, -3) instead of (-3, -3, -2), leading to an incorrect dot product and magnitude calculation.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution computes the sine of the angle using a cross-product-like numerator instead of the dot product required for the cosine formula. The correct dot product is 17, not 3.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.