Planes: distances and angles
Problem 9.816 · medium
Find the angle between the planes \( \displaystyle x + 4 y + 4 z = -3 \) and \( \displaystyle - 2 x - y - 2 z = 4 \).
- The angle between planes is the (acute) angle between their normals.
- \[ \frac{\left|{4 \left(-2\right) + 4 \left(-1\right) + 1 \left(-2\right)}\right|}{\sqrt{\left(-1\right)^{2} + \left(-2\right)^{2} + \left(-2\right)^{2}} \sqrt{1^{2} + 4^{2} + 4^{2}}} = \frac{14 \sqrt{33}}{99} \]cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
- θ = arccos of that: acos(14*sqrt(33)/99).
Answer \( \operatorname{acos}{\left(\frac{14 \sqrt{33}}{99} \right)} \approx 35.67^\circ \)
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: passqwen3.6:27b-mlx: fail (error) — The normal vector for the first plane x + 4y + 4z = -3 is (1, 4, 4), but the calculation in line 2 uses (4, 4, 1). This incorrect normal vector leads to a wrong dot product and magnitude, resulting in an incorrect angle.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The normal vector for the first plane x + 4y + 4z = -3 is (1, 4, 4), but the calculation in line 2 uses (4, 4, 1). This incorrect normal vector leads to a wrong dot product and magnitude, resulting in an incorrect angle.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the angle between planes as the acute angle between their normal vectors. The calculation of the dot product and magnitudes is correct, and the final result matches the stated answer.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-18 — The normal vector for the first plane is incorrectly identified as (-1, -2, -2) in the calculation, whereas the equation x + 4y + 4z = -3 implies the normal is (1, 4, 4). The second normal is also swapped or misassigned relative to the coefficients. Specifically, the dot product calculation uses components that do not match the given plane equations.gpt-oss:20b: pass 2026-10-18
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 2026-10-19 with SymPy 1.14.0.