Home›Calculus 3›Planes: distances and angles›Problem 9.774

Planes: distances and angles

Problem 9.774 · easy

Find an equation of the plane through \( \displaystyle (-1, 1, 1) \) with normal vector \( \displaystyle \left\langle 3, -1, 4 \right\rangle \).

n · (r − P) = 0, i.e. ax + by + cz = n · P.
Reviewed
\[ 3 \left(-1\right) - 1 \cdot 1 + 4 \cdot 1 = 0 \]
n · P.✓ Proved

Answer \( 3 x - y + 4 z = 0 \)

✓ 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 plane holds P and its gradient is parallel to n

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the point-normal form of a plane equation. The calculation of the constant term is correct, and the final equation matches the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies the point-normal form of a plane equation. The calculation of the constant term is correct, and the final equation matches the stated answer.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the point-normal form of the plane equation. The calculation of the constant term is correct, and the final equation matches the stated answer.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the point-normal form of a plane equation. The calculation of the constant term d = n · P is correct, and the final equation matches the stated answer.
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.