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Planes: distances and angles

Problem 9.634 · medium

Find the angle between the planes \( \displaystyle - x - 2 y - 2 z = 5 \) and \( \displaystyle 3 x + y + 2 z = 3 \).

The angle between planes is the (acute) angle between their normals.
\[ \frac{\left|{\left(-2\right) 2 - 1 \cdot 3 - 2 \cdot 1}\right|}{\sqrt{\left(-1\right)^{2} + \left(-2\right)^{2} + \left(-2\right)^{2}} \sqrt{1^{2} + 2^{2} + 3^{2}}} = \frac{3 \sqrt{14}}{14} \]
cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
θ = arccos of that: acos(3*sqrt(14)/14).

Answer \( \operatorname{acos}{\left(\frac{3 \sqrt{14}}{14} \right)} \approx 36.70^\circ \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution computes the cosine of the angle using the normal vector <1, 2, 3> for the second plane, but the equation 3x + y + 2z = 3 has normal vector <3, 1, 2>. This incorrect normal vector leads to the wrong numerical answer.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution computes the cosine of the angle using the normal vector <1, 2, 3> for the second plane, but the equation 3x + y + 2z = 3 has normal vector <3, 1, 2>. This incorrect normal vector leads to the wrong numerical answer.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution computes the cosine of the angle using incorrect normal vectors. The normal for the second plane 3x + y + 2z = 3 is <3, 1, 2>, but the calculation uses <1, 2, 3> (evident from the denominator term sqrt(1**2 + 2**2 + 3**2) and the dot product terms). This leads to an incorrect final answer.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution uses incorrect normal vectors for the second plane. The plane 3x + y + 2z = 3 has normal vector <3, 1, 2>, but the calculation uses <1, 2, 3>. This leads to an incorrect angle.
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.