Home›Calculus 3›Planes: distances and angles›Problem 9.627

Planes: distances and angles

Problem 9.627 · easy

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

Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖.
Reviewed
\[ \frac{\left|{\left(-4\right) 3 + 2 \left(-2\right) + 1 \cdot 3 + 4}\right|}{\sqrt{1^{2} + 2^{2} + \left(-4\right)^{2}}} = \frac{3 \sqrt{21}}{7} \]
Substitute.✓ Proved

Answer \( \frac{3 \sqrt{21}}{7} \)

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 plane equation is implicitly treated as -4x + 2y + z + 4 = 0 (so d = -4 in the standard ax+by+cz=d form, or constant term 4 in the numerator calculation), and the normal vector magnitude is calculated correctly.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the point-to-plane distance formula. The plane equation is implicitly treated as -4x + 2y + z + 4 = 0 (so d = -4 in the standard ax+by+cz=d form, or constant term 4 in the numerator calculation), and the normal vector magnitude is calculated correctly.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the distance formula and substitutes the coefficients and point coordinates accurately. The algebraic simplification is correct.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — 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) 2026-10-14 — The sentence states the distance formula with a minus sign before d, but the correct formula uses +d when the plane is written as ax+by+cz+d=0. The subsequent calculation uses the correct sign, so the sentence is misleading.

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.