Planes: distances and angles
Problem 9.814 · medium
Find the angle between the planes \( \displaystyle x + 2 y - 4 z = 1 \) and \( \displaystyle 2 x - y - 2 z = 1 \).
- The angle between planes is the (acute) angle between their normals.
- \[ \frac{\left|{2 \left(-1\right) + 1 \cdot 2 - 4 \left(-2\right)}\right|}{\sqrt{\left(-1\right)^{2} + 2^{2} + \left(-2\right)^{2}} \sqrt{1^{2} + 2^{2} + \left(-4\right)^{2}}} = \frac{8 \sqrt{21}}{63} \]cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
- θ = arccos of that: acos(8*sqrt(21)/63).
Answer \( \operatorname{acos}{\left(\frac{8 \sqrt{21}}{63} \right)} \approx 54.41^\circ \)
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 solution uses incorrect normal vectors for the given planes. The plane x + 2y - 4z = 1 has normal <1, 2, -4>, but the calculation uses <-1, 2, -2> (or similar incorrect components). Similarly, the second plane 2x - y - 2z = 1 has normal <2, -1, -2>, but the calculation uses <1, 2, -4> mixed with other errors. Specifically, the dot product calculation `2*(-1) + 1*2 - 4*(-2)` implies normals like <-1, 1, -4> and <2, 2, -2> or some other mismatch that does not correspond to the problem statement.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution uses incorrect normal vectors for the given planes. The plane x + 2y - 4z = 1 has normal <1, 2, -4>, but the calculation uses <-1, 2, -2> (or similar incorrect components). Similarly, the second plane 2x - y - 2z = 1 has normal <2, -1, -2>, but the calculation uses <1, 2, -4> mixed with other errors. Specifically, the dot product calculation `2*(-1) + 1*2 - 4*(-2)` implies normals like <-1, 1, -4> and <2, 2, -2> or some other mismatch that does not correspond to the problem statement.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution uses incorrect normal vectors for the given planes. The plane x + 2y - 4z = 1 has normal (1, 2, -4), but the calculation uses (-1, 2, -2). The plane 2x - y - 2z = 1 has normal (2, -1, -2), but the calculation uses (1, 2, -4). The vectors used in the dot product and norms do not correspond to the planes in the problem statement.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer returned a non-objectgpt-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.