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Equation of a plane

Problem 9.81 · medium

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

\[ \left[\begin{matrix}0\\5\\2\end{matrix}\right] \]
PQ.Reviewed
\[ \left[\begin{matrix}0\\3\\2\end{matrix}\right] \]
PR.Reviewed
\[ \left[\begin{matrix}1\\0\\0\end{matrix}\right] \]
A normal vector n = PQ × PR (scaled).Reviewed
\[ -1 \]
n · P gives the constant.Reviewed
The plane is x = -1.
Reviewed

Answer \( x = -1 \)

✓ Nihil obstat Lines: 5 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	both sides are the same expression: the line states a value, and the working that produced it is not on the line
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
4	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
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 that all three points share the same x-coordinate, leading to the plane equation x = -1. The vector calculations and normal vector derivation are consistent with this result.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies that all three points share the same x-coordinate, leading to the plane equation x = -1. The vector calculations and normal vector derivation are consistent with this result.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies that all points share the same x-coordinate, leading to the plane equation x = -1. The intermediate vector calculations are consistent with this result.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies that all three points have an x-coordinate of -1, implying the plane is x = -1. The vector calculations confirm the normal vector is parallel to the x-axis.
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.