Planes: distances and angles
Problem 9.714 · medium
Find the angle between the planes \( \displaystyle 4 x - 4 y - 2 z = -5 \) and \( \displaystyle x - 2 y - 3 z = -2 \).
- The angle between planes is the (acute) angle between their normals.Reviewed
- \[ \frac{\left|{4 \cdot 1 - 2 \left(-3\right) - 4 \left(-2\right)}\right|}{\sqrt{\left(-2\right)^{2} + 4^{2} + \left(-4\right)^{2}} \sqrt{1^{2} + \left(-2\right)^{2} + \left(-3\right)^{2}}} = \frac{3 \sqrt{14}}{14} \]cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
- θ = arccos of that: acos(3*sqrt(14)/14).Reviewed
Answer \( \operatorname{acos}{\left(\frac{3 \sqrt{14}}{14} \right)} \approx 36.70^\circ \)
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: passqwen3.6:27b-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 computes the result accurately.
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 computes the result accurately.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The normal vector for the first plane is incorrectly identified as (-2, 4, -4) instead of (4, -4, -2). While the dot product magnitude happens to be the same, the norm calculation uses the wrong vector components, leading to an incorrect denominator and final answer.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The normal vector for the first plane is incorrectly identified as (-2, 4, -4) instead of (4, -4, -2). This error happens to produce the correct magnitude but yields an incorrect dot product calculation in the numerator, although the final numerical result coincidentally matches the correct answer due to a calculation error in the provided equation line (4*1 - 2*(-3) - 4*(-2) = 14, not 9*sqrt(14)/sqrt(14)*sqrt(14) etc. Wait, let's re-evaluate. Correct normals: n1=(4,-4,-2), n2=(1,-2,-3). Dot product: 4 + 8 + 6 = 18. |n1|=sqrt(16+16+4)=6. |n2|=sqrt(1+4+9)=sqrt(14). Cos theta = 18/(6*sqrt(14)) = 3/sqrt(14) = 3*sqrt(14)/14. The solution's equation uses vectors (-2, 4, -4) and (1, -2, -3). Dot product: -2 -8 + 8 = -2. Abs(-2) = 2. This would give 2/(6*sqrt(14)) = 1/(3*sqrt(14)). The equation line claims the result is 3*sqrt(14)/14. The equation line is mathematically false: LHS is 2/(6*sqrt(14)) which is not equal to RHS. The setup sentence is correct, but the equation is wrong and the derivation is inconsistent.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.