Planes: distances and angles
Problem 9.722 · easy
Find the distance from \( \displaystyle (-3, 4, -4) \) to the plane \( \displaystyle 4 x + 2 y - 3 z = 4 \).
- Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖.
- \[ \frac{\left|{4 \left(-3\right) - 4 + 2 \cdot 4 - 3 \left(-4\right)}\right|}{\sqrt{2^{2} + \left(-3\right)^{2} + 4^{2}}} = \frac{4 \sqrt{29}}{29} \]Substitute.✓ Proved
Answer \( \frac{4 \sqrt{29}}{29} \)
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: passqwen3.6:27b-mlx: fail (error) — The formula in line 1 is incorrect; it includes an extra term 'ax₀' inside the absolute value, leading to an incorrect substitution in line 2. The correct numerator should be |4(-3) + 2(4) - 3(-4) - 4|.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The formula in line 1 is incorrect; it includes an extra term 'ax₀' inside the absolute value, leading to an incorrect substitution in line 2. The correct numerator should be |4(-3) + 2(4) - 3(-4) - 4|.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution incorrectly identifies the constant term d as 4 instead of -4 in the distance formula, leading to an incorrect numerator calculation. The formula |ax₀ + by₀ + cz₀ - d| requires d to be the constant on the RHS of the plane equation, so it should be |-4 - 4| = 8, not |-4 + ...|.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The substitution in line 2 is incorrect: it uses coefficients (4, 2, -3) for the plane but calculates the norm using (2, -3, 4). Additionally, the numerator calculation is garbled and does not match the standard formula application.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.