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Planes: distances and angles

Problem 9.725 · easy

Find the distance from \( \displaystyle (-3, 3, 2) \) to the plane \( \displaystyle 4 x + 3 y + 3 z = -6 \).

Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖.
Reviewed
\[ \frac{\left|{4 \left(-3\right) + 3 \cdot 2 + 6 + 3 \cdot 3}\right|}{\sqrt{3^{2} + 3^{2} + 4^{2}}} = \frac{9 \sqrt{34}}{34} \]
Substitute.✓ Proved

Answer \( \frac{9 \sqrt{34}}{34} \)

Lines: 1 proved, 1 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
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: pass
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 simplification 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 simplification is correct.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the point-to-plane distance formula. The plane equation is implicitly treated as 4x + 3y + 3z + 6 = 0 (so d = -6 in the standard form ax+by+cz+d=0, or equivalently the constant term is moved to the RHS), and the substitution of coordinates and normal vector magnitude is correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The setup formula uses 'd' as the constant on the RHS of the plane equation (4x+3y+3z=d), but the substitution incorrectly adds 6 instead of subtrac
gpt-oss:20b: fail (misleading) 2026-10-15 — The sentence states the distance formula as |ax₀+by₀+cz₀−d|/‖n‖, which is correct only when the plane is written as ax+by+cz−d=0. Here the plane is 4x+3y+3z=−6, i.e. 4x+3y+3z+6=0, so the correct formula is |ax₀+by₀+cz₀+6|/‖n‖. The solution’s calculation uses +6, but the sentence misstates the sign, potentially confusing students about the proper form of the plane equation and the distance formula.

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.