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Equation of a plane

Problem 9.40 · medium

Find an equation of the plane through the points \( \displaystyle (-2, -1, -2),\ (2, 0, 3),\ (0, 0, 1) \).

\[ \left[\begin{matrix}4\\1\\5\end{matrix}\right] \]
PQ.
\[ \left[\begin{matrix}2\\1\\3\end{matrix}\right] \]
PR.
\[ \left[\begin{matrix}-1\\-1\\1\end{matrix}\right] \]
A normal vector n = PQ × PR (scaled).
\[ 1 \]
n · P gives the constant.
The plane is -x - y + z = 1.

Answer \( - x - y + z = 1 \)

Lines: 5 not checked. 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	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
2	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
3	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
4	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
5	Not checked	—	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: inconclusive — reviewer returned a non-object

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer returned a non-object
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: inconclusive 2026-09-27 — reviewer returned a non-object
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The normal vector calculation is incorrect. The cross product of vectors PQ=<4,1,5> and PR=<2,1,3> is <-2, -2, 2>, which simplifies to <-1, -1, 1>. H
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 2026-10-13 with SymPy 1.14.0.