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Planes: distances and angles

Problem 9.630 · medium

Find the angle between the planes \( \displaystyle 3 x - y + 4 z = 0 \) and \( \displaystyle - x + y + z = -1 \).

The angle between planes is the (acute) angle between their normals.
Reviewed
\[ \frac{\left|{3 \left(-1\right) - 1 \cdot 1 + 4 \cdot 1}\right|}{\sqrt{\left(-1\right)^{2} + 1^{2} + 1^{2}} \sqrt{\left(-1\right)^{2} + 3^{2} + 4^{2}}} = 0 \]
cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
\[ \operatorname{acos}{\left(0 \right)} = \frac{\pi}{2} \]
θ = arccos of that.✓ Proved

Answer \( \frac{\pi}{2} \approx 90.00^\circ \)

✓ Nihil obstat Lines: 2 proved, 1 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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
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, computes the dot product as zero, and concludes the planes are orthogonal. The logic is sound.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the normals, computes the dot product as zero, and concludes the planes are orthogonal. The logic is sound.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the normals, applies the formula for the angle between planes, and computes the result accurately.
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, 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.