Home›Calculus 3›Planes: distances and angles›Problem 9.818

Planes: distances and angles

Problem 9.818 · easy

Find the distance from \( \displaystyle (1, -1, -3) \) to the plane \( \displaystyle 2 x + 2 y + 2 z = 6 \).

Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖.
Reviewed
\[ \frac{\left|{2 \left(-3\right) - 6 + 2 \left(-1\right) + 2 \cdot 1}\right|}{\sqrt{2^{2} + 2^{2} + 2^{2}}} = 2 \sqrt{3} \]
Substitute.✓ Proved

Answer \( 2 \sqrt{3} \)

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 normal vector components is accurate, and the algebraic simplification is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies the point-to-plane distance formula. The substitution of coordinates and normal vector components is accurate, and the algebraic simplification is correct.
gpt-oss:20b: pass 2026-10-19
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: fail (misleading) 2026-10-18 — The sentence claims the distance formula is applied correctly, but the substitution swaps the coordinates of the point (1,‑1,‑3) to (‑3,‑1,1). Although the numerical result coincidentally matches, the setup is incorrect and could mislead a student about the proper use of the formula.
qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The formula used in step 1 is incorrect; it should be |ax₀ + by₀ + cz₀ - d| / ||n||, but the numerator in the standard formula is ax₀ + by₀ + cz₀ -
gpt-oss:20b: pass 2026-10-18

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.