Planes: distances and angles
Problem 9.767 · easy
Find the distance from \( \displaystyle (0, 2, -1) \) to the plane \( \displaystyle 2 x + y - z = -1 \).
- Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖.
- \[ \frac{\left|{2 \cdot 0 + 1 - 1 \left(-1\right) + 1 \cdot 2}\right|}{\sqrt{1^{2} + \left(-1\right)^{2} + 2^{2}}} = \frac{2 \sqrt{6}}{3} \]Substitute.✓ Proved
Answer \( \frac{2 \sqrt{6}}{3} \)
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 plane was rewritten incorrectly: 2x+y−z=−1 should be 2x+y−z+1=0, so d=1. The distance formula then gives |2|/√6=√6/3, not 2√6/3. The solution’s algebraic substitution is based on a wrong d and yields an incorrect result.qwen3.6:27b-mlx: fail (error) — The formula in step 1 is inconsistent with the substitution in step 2. Step 1 defines the denominator as the norm of n, but step 2 uses coefficients (1, -1, 2) which do not match the plane equation 2x + y - z = -1 (coefficients 2, 1, -1). Additionally, the numerator calculation in step 2 is arithmetically incorrect for the stated coefficients.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The formula in step 1 is inconsistent with the substitution in step 2. Step 1 defines the denominator as the norm of n, but step 2 uses coefficients (1, -1, 2) which do not match the plane equation 2x + y - z = -1 (coefficients 2, 1, -1). Additionally, the numerator calculation in step 2 is arithmetically incorrect for the stated coefficients.gpt-oss:20b: fail (error) 2026-10-19 — The plane was rewritten incorrectly: 2x+y−z=−1 should be 2x+y−z+1=0, so d=1. The distance formula then gives |2|/√6=√6/3, not 2√6/3. The solution’s algebraic substitution is based on a wrong d and yields an incorrect result.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The formula in step 1 uses 'd' as the constant term on the RHS, but step 2 substitutes d=1 for the equation 2x+y-z=-1 (where d should be -1). Additionally, the coefficients a, b, c in step 2 (1, -1, 2) do not match the plane equation 2x+y-z=-1 (2, 1, -1).gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The formula in step 1 uses 'd' as the constant term on the right-hand side of the plane equation, but the substitution in step 2 incorrectly adds 1gpt-oss:20b: fail (error) 2026-10-15 — The solution misidentifies the plane’s normal vector and the constant term. Using the correct normal (2,1,−1) and d=−1 gives distance |2·0+1·2−(−1)·(−1)+1|/√(2²+1²+(-1)²)=2/√6=√6/3, not 2√6/3.
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.