Planes: distances and angles
Problem 9.770 · medium
Find the angle between the planes \( \displaystyle x + 4 y - z = -5 \) and \( \displaystyle - 3 x - y + 3 z = -2 \).
- The angle between planes is the (acute) angle between their normals.
- \[ \frac{\left|{4 \left(-1\right) + 1 \left(-3\right) - 1 \cdot 3}\right|}{\sqrt{\left(-1\right)^{2} + \left(-3\right)^{2} + 3^{2}} \sqrt{1^{2} + \left(-1\right)^{2} + 4^{2}}} = \frac{5 \sqrt{38}}{57} \]cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
- θ = arccos of that: acos(5*sqrt(38)/57).
Answer \( \operatorname{acos}{\left(\frac{5 \sqrt{38}}{57} \right)} \approx 57.27^\circ \)
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 first plane is incorrectly identified as <1, -1, 4> instead of <1, 4, -1>. This leads to an incorrect dot product and magnitude, resulting in the wrong angle.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The normal vector for the first plane is incorrectly identified as <1, -1, 4> instead of <1, 4, -1>. This leads to an incorrect dot product and magnitude, resulting in the wrong angle.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-16gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The normal vectors extracted in the calculation (n1 = <-1, -3, 3> and n2 = <1, -1, 4>) do not correspond to the planes given in the problem statement (n1 should be <1, 4, -1> and n2 should be <-3, -1, 3>). Although the final numerical answer happens to be correct due to a coincidence in magnitudes and dot products, the solution demonstrates incorrect extraction of normal vectors.gpt-oss:20b: pass 2026-10-15
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.