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Planes: distances and angles

Problem 9.635 · medium

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

The angle between planes is the (acute) angle between their normals.
Reviewed
\[ \frac{\left|{\left(-3\right) 3 + 3 \left(-1\right) + 3 \cdot 2}\right|}{\sqrt{\left(-1\right)^{2} + 2^{2} + 3^{2}} \sqrt{3^{2} + \left(-3\right)^{2} + 3^{2}}} = \frac{\sqrt{42}}{21} \]
cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
θ = arccos of that: acos(sqrt(42)/21).
Reviewed

Answer \( \operatorname{acos}{\left(\frac{\sqrt{42}}{21} \right)} \approx 72.02^\circ \)

✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 angle between planes as the acute angle between their normal vectors and applies the standard formula involving the absolute value of the dot product. The calculation and final result are correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the angle between planes as the acute angle between their normal vectors and applies the standard formula involving the absolute value of the dot product. The calculation and final result are correct.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the angle between planes as the acute angle between their normal vectors and applies the standard formula involving the absolute value of the dot product. The calculation is correct.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — 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 calculates the result accurately.
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.