Planes: distances and angles
Problem 9.632 · easy
Find the distance from \( \displaystyle (0, -3, 4) \) to the plane \( \displaystyle 3 x + 3 y + 2 z = -1 \).
- Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖.
- \[ \frac{\left|{3 \left(-3\right) + 3 \cdot 0 + 1 + 2 \cdot 4}\right|}{\sqrt{2^{2} + 3^{2} + 3^{2}}} = 0 \]Substitute.✓ Proved
Answer \( 0 \)
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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The formula stated in line 1 is incorrect; it should be |ax₀ + by₀ + cz₀ + d| / ||n|| for a plane ax + by + cz + d = 0, or |ax₀ + by₀ + cz₀ - d| / |
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-17 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The formula stated in line 1 is incorrect; it should be |ax₀ + by₀ + cz₀ + d| / ||n|| for a plane ax + by + cz + d = 0, or |ax₀ + by₀ + cz₀ - d| / |gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-15 — The formula in line 1 is incorrect; it should be |ax₀ + by₀ + cz₀ + d| / ‖n‖ for a plane ax + by + cz = -d, or |ax₀ + by₀ + cz₀ - D| / ‖n‖ for ax + by + cz = D. The solution incorrectly adds 1 instead of subtracting -1 (or adding 1 if the constant term was on the RHS as -1, but the standard form usually implies moving everything to one side). Specifically, for 3x+3y+2z = -1, the distance is |3(0)+3(-3)+2(4) - (-1)| / sqrt(22) = |-9+8+1|/sqrt(22) = 0. Wait, let's re-evaluate. Point (0, -3, 4). 3(0) + 3(-3) + 2(4) = -9 + 8 = -1. The plane equation is 3x+3y+2z = -1. So the point satisfies the equation. The distance is indeed 0. However, the formula stated in line 1 is `|ax₀ + by₀ + cz₀ − d|`. If the plane is `ax+by+cz=d`, then the formula is correct. Here the plane is `3x+3y+2z = -1`, so `d = -1`. The formula becomes `|... - (-1)| = |... + 1|`. Line 2 calculates `Abs((3*(-3) + 3*0 + 2*4) + 1)`. This matches `| -9 + 8 + 1 | = 0`. So the calculation is correct. Is the formula statement misleading? It says `Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖`. This is the standard formula for plane `ax+by+cz=d`. The problem gives `3x+3y+2z = -1`. So `d=-1`. The substitution in line 2 uses `+1`, which corresponds to `-d` where `d=-1`. So the logic holds. Why did I think it was wrong? Let's look closer. Line 1: `Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖`. This is a standard template. Line 2: `Abs((3*(-3) + 3*0 + 2*4) + 1)`. The term `+1` comes from `-d` if `d=-1`. The plane is `3x+3y+2z = -1`. So `d=-1`. `-d = 1`. So `+1` is correct. The result is 0. The point is on the plane. The solution is correct.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The formula in step 1 is incorrect; it should be |ax₀ + by₀ + cz₀ + d| / ‖n‖ for a plane ax + by + cz = -d, or |ax₀ + by₀ + cz₀ - d| / ‖n‖ for ax + by + cz = d. The solution mixes the sign convention, leading to an incorrect numerator calculation (it adds 1 instead of subtracting -1 or handling the constant term correctly relative to the standard form).gpt-oss:20b: pass 2026-10-14
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 2026-10-17 with SymPy 1.14.0.