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Equation of a plane

Problem 9.113 · medium

Find an equation of the plane through the points \( \displaystyle (-1, -1, -2),\ (-2, 0, -2),\ (-1, -1, 1) \).

\[ \left[\begin{matrix}-1\\1\\0\end{matrix}\right] \]
PQ.✓ Proved
\[ \left[\begin{matrix}0\\0\\3\end{matrix}\right] \]
PR.✓ Proved
\[ \left[\begin{matrix}1\\1\\0\end{matrix}\right] \]
A normal vector n = PQ × PR (scaled).✓ Proved
\[ -2 \]
n · P gives the constant.✓ Proved
The plane is x + y = -2.
Reviewed

Answer \( x + y = -2 \)

Lines: 4 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.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
5	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	all three points satisfy the equation

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the normal vector and the constant term, leading to the correct equation of the plane.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the normal vector and the constant term, leading to the correct equation of the plane.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer returned a non-object
gpt-oss:20b: pass 2026-09-26

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/plane_through_points, checked 2026-09-26 with SymPy 1.14.0.