Home›Calculus 3›Planes: distances and angles›Problem 9.776

Planes: distances and angles

Problem 9.776 · easy

Find the distance from \( \displaystyle (4, -2, 0) \) to the plane \( \displaystyle 4 x + 4 y + 2 z = 5 \).

Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖.
Reviewed
\[ \frac{\left|{4 \left(-2\right) - 5 + 2 \cdot 0 + 4 \cdot 4}\right|}{\sqrt{2^{2} + 4^{2} + 4^{2}}} = \frac{1}{2} \]
Substitute.✓ Proved

Answer \( \frac{1}{2} \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 arithmetic yields the correct result.

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 arithmetic yields the correct result.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the point-to-plane distance formula. The substitution of coordinates and normal vector components is accurate, and the arithmetic yields the correct result.
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 2026-10-19 with SymPy 1.14.0.