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Equation of a plane

Problem 9.82 · medium

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

\[ \left[\begin{matrix}3\\-3\\1\end{matrix}\right] \]
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: fail (error) — The cross product of PQ and PR was computed incorrectly; the normal vector should be proportional to (-7,7,0), not (-1,-1,0). Consequently the derived plane equation -x - y = -2 is wrong. The correct plane is y - x = 2.
qwen3.6:27b-mlx: pass — The solution correctly identifies two vectors in the plane, computes a valid normal vector via cross product, and derives the correct plane equation. The final equation -x - y = -2 is equivalent to x + y = 2, which satisfies all three given points.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies two vectors in the plane, computes a valid normal vector via cross product, and derives the correct plane equation. The final equation -x - y = -2 is equivalent to x + y = 2, which satisfies all three given points.
gpt-oss:20b: fail (error) 2026-09-28 — The cross product of PQ and PR was computed incorrectly; the normal vector should be proportional to (-7,7,0), not (-1,-1,0). Consequently the derived plane equation -x - y = -2 is wrong. The correct plane is y - x = 2.
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies vectors lying in the plane, computes a valid normal vector, and derives the correct plane equation. The final answer matches the stated answer.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies two vectors in the plane, computes a normal vector via cross product, and determines the plane equation. The final equation -x - y = -2 is correct for the given points.
gpt-oss:20b: fail (error) 2026-09-26 — The cross product used to find the normal vector is incorrect, leading to an incorrect plane equation that does not contain all three given points.

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.