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Equation of a plane

Problem 9.114 · medium

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

\[ \left[\begin{matrix}0\\0\\-5\end{matrix}\right] \]
PQ.
\[ \left[\begin{matrix}1\\1\\-2\end{matrix}\right] \]
PR.
\[ \left[\begin{matrix}1\\-1\\0\end{matrix}\right] \]
A normal vector n = PQ × PR (scaled).
\[ -2 \]
n · P gives the constant.
The plane is x - y = -2.

Answer \( x - y = -2 \)

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 response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The vectors PQ and PR are calculated incorrectly. Given points A(-1,1,3), B(-1,1,-2), C(0,2,1), vector AB is <0,0,-5> and vector AC is <1,1,-2>. The

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-09-27 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The vectors PQ and PR are calculated incorrectly. Given points A(-1,1,3), B(-1,1,-2), C(0,2,1), vector AB is <0,0,-5> and vector AC is <1,1,-2>. The
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer returned a non-object
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The vectors PQ and PR are calculated incorrectly. P(-1,1,3), Q(-1,1,-2), R(0,2,1). PQ should be <0, 0, -5> and PR should be <1, 1, -2>. The solution
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.