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Equation of a plane

Problem 9.208 · hard

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

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

Answer \( 3 x + y - z = -2 \)

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

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-03
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies the normal vector via the cross product of displacement vectors and verifies the constant term using one of the given points. The final equation is correct.
gpt-oss:20b: fail (error) 2026-10-03 — The cross product PQ×PR is computed incorrectly; the resulting normal vector (3,1,-1) is not orthogonal to both PQ and PR. Consequently the derived plane equation 3x+y−z=−2 does not pass through the 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 2026-10-03 with SymPy 1.14.0.