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Planes: distances and angles

Problem 9.636 · medium

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

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

Answer \( \operatorname{acos}{\left(\frac{2 \sqrt{2}}{3} \right)} \approx 19.47^\circ \)

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 normal vectors are incorrectly identified. For x - 2y + z = 2, n1 = <1, -2, 1>, but the calculation uses <1, 1, -2>. For x - y + z = -5, n2 = <1, -1, 1>, but the calculation uses <1, -1, 1> (this one is correct, but the first is wrong). The dot product and magnitudes are calculated based on these wrong vectors.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The normal vectors are incorrectly identified. For x - 2y + z = 2, n1 = <1, -2, 1>, but the calculation uses <1, 1, -2>. For x - y + z = -5, n2 = <1, -1, 1>, but the calculation uses <1, -1, 1> (this one is correct, but the first is wrong). The dot product and magnitudes are calculated based on these wrong vectors.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The normal vectors are incorrectly identified. For x - 2y + z = 2, n1 = <1, -2, 1>, but the calculation uses <1, 1, -2>. For x - y + z = -5, n2 = <1, -1, 1>, but the calculation uses <1, -1, 1> (this one is correct, but the first is wrong). The dot product and magnitudes in line 2 do not match the actual normal vectors of the given planes.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The normal vectors are incorrectly identified. The plane x - 2y + z = 2 has normal <1, -2, 1>, but the calculation uses <1, 1, -2>. Similarly, x - y + z = -5 has normal <1, -1, 1>, but the calculation uses <1, -1, 1> (this one is correct, but the first is wrong). The dot product and magnitudes are calculated based on these wrong normals.
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.