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Planes: distances and angles

Problem 9.718 · medium

Find the angle between the planes \( \displaystyle 4 x - 3 y + 3 z = 3 \) and \( \displaystyle 2 x - 2 y - 3 z = 1 \).

The angle between planes is the (acute) angle between their normals.
Reviewed
\[ \frac{\left|{3 \left(-3\right) - 3 \left(-2\right) + 4 \cdot 2}\right|}{\sqrt{\left(-3\right)^{2} + 3^{2} + 4^{2}} \sqrt{2^{2} + \left(-2\right)^{2} + \left(-3\right)^{2}}} = \frac{5 \sqrt{2}}{34} \]
cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
θ = arccos of that: acos(5*sqrt(2)/34).
Reviewed

Answer \( \operatorname{acos}{\left(\frac{5 \sqrt{2}}{34} \right)} \approx 78.00^\circ \)

Lines: 1 proved, 2 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly identifies the normals, applies the formula for the angle between planes using the absolute value of the dot product to ensure an acute angle, and arrives at the correct result.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the normals, applies the formula for the angle between planes using the absolute value of the dot product to ensure an acute angle, and arrives at the correct result.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution computes the cosine of the angle between the normal vectors, but fails to take the absolute value of the dot product in the numerator.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the normals, applies the formula for the angle between planes using the absolute value of the dot product to ensure the acute angle, and computes the result accurately.
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 2026-10-18 with SymPy 1.14.0.