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Planes: distances and angles

Problem 9.637 · medium

Find the angle between the planes \( \displaystyle - 4 x + 2 y + z = 2 \) and \( \displaystyle - 2 x - y + z = 5 \).

The angle between planes is the (acute) angle between their normals.
\[ \frac{\left|{2 \left(-1\right) + 1 \cdot 1 - 4 \left(-2\right)}\right|}{\sqrt{\left(-1\right)^{2} + 1^{2} + \left(-2\right)^{2}} \sqrt{1^{2} + 2^{2} + \left(-4\right)^{2}}} = \frac{\sqrt{14}}{6} \]
cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
θ = arccos of that: acos(sqrt(14)/6).

Answer \( \operatorname{acos}{\left(\frac{\sqrt{14}}{6} \right)} \approx 51.42^\circ \)

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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The normals used in the dot‑product calculation are incorrect. For the planes \(-4x+2y+z=2\) and \(-2x-y+z=5\) the normals are \((-4,2,1)\) and \((-2,-1,1
qwen3.6:27b-mlx: fail (error) — The normal vectors used in the calculation do not match the planes given in the problem statement. The equation uses components like (-1, 1, -2) and (1, 2, -4), but the planes -4x + 2y + z = 2 and -2x - y + z = 5 have normals (-4, 2, 1) and (-2, -1, 1).

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The normal vectors used in the calculation do not match the planes given in the problem statement. The equation uses components like (-1, 1, -2) and (1, 2, -4), but the planes -4x + 2y + z = 2 and -2x - y + z = 5 have normals (-4, 2, 1) and (-2, -1, 1).
gpt-oss:20b: inconclusive 2026-10-17 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The normals used in the dot‑product calculation are incorrect. For the planes \(-4x+2y+z=2\) and \(-2x-y+z=5\) the normals are \((-4,2,1)\) and \((-2,-1,1
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The normal vectors extracted from the plane equations are incorrect. For -4x + 2y + z = 2, the normal is <-4, 2, 1>, but the solution uses <-1, 1, -2>. For -2x - y + z = 5, the normal is <-2, -1, 1>, but the solution uses <1, 2, -4>. The algebraic computation in step 2 does not match the normals derived from the problem statement.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The normal vectors extracted from the plane equations are incorrect. For -4x + 2y + z = 2, the normal is <-4, 2, 1>, but the calculation uses <-1, 1, -2>. For -2x - y + z = 5, the normal is <-2, -1, 1>, but the calculation uses <1, 2, -4>. The setup does not model the problem as stated.
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.