Planes: distances and angles
Problem 9.724 · easy
Find the distance from \( \displaystyle (1, -1, 2) \) to the plane \( \displaystyle - 2 x + 4 y + z = -1 \).
- Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖.
- \[ \frac{\left|{4 \left(-1\right) - 2 \cdot 1 + 1 + 1 \cdot 2}\right|}{\sqrt{1^{2} + \left(-2\right)^{2} + 4^{2}}} = \frac{\sqrt{21}}{7} \]Substitute.✓ Proved
Answer \( \frac{\sqrt{21}}{7} \)
Lines: 1 proved, 1 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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the foot of the perpendicular found numerically, then the distance to it |
Reviewers
gpt-oss:20b: fail (error) — The sentence correctly states the distance formula, but the equation substitutes the coefficients of a different plane (4x‑2y+z=1) instead of the given plane –2x+4y+z=−1. The calculation is not justified by the stated problem, so the solution is incorrect.qwen3.6:27b-mlx: pass — The solution correctly applies the point-to-plane distance formula. The substitution of coordinates and plane coefficients is accurate, and the final arithmetic is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the point-to-plane distance formula. The substitution of coordinates and plane coefficients is accurate, and the final arithmetic is correct.gpt-oss:20b: fail (error) 2026-10-18 — The sentence correctly states the distance formula, but the equation substitutes the coefficients of a different plane (4x‑2y+z=1) instead of the given plane –2x+4y+z=−1. The calculation is not justified by the stated problem, so the solution is incorrect.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the point-to-plane distance formula. The substitution of coordinates and plane coefficients is accurate, and the final arithmetic is correct.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The formula in step 1 is ambiguous regarding the sign of d, leading to an incorrect substitution in step 2. For the plane -2x + 4y + z = -1, the numerator should be |-2(1) + 4(-1) + 1(2) - (-1)| = |-3| = 3, not 1. The solution incorrectly adds 1 instead of subtracting -1 (or equivalently, uses the wrong sign for the constant term relative to the normal vector components).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.