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Equation of a plane

Problem 9.184 · hard

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

\[ \left[\begin{matrix}-2\\1\\3\end{matrix}\right] \]
PQ.✓ Proved
\[ \left[\begin{matrix}3\\-1\\0\end{matrix}\right] \]
PR.✓ Proved
\[ \left[\begin{matrix}3\\9\\-1\end{matrix}\right] \]
A normal vector n = PQ × PR (scaled).✓ Proved
\[ 16 \]
n · P gives the constant.✓ Proved
The plane is 3*x + 9*y - z = 16.
Reviewed

Answer \( 3 x + 9 y - z = 16 \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

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 — The solution correctly identifies two vectors lying on the plane, computes their cross product to find a normal vector, and determines the constant term using one of the given points. The final equation is correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies two vectors lying on the plane, computes their cross product to find a normal vector, and determines the constant term using one of the given points. The final equation is correct.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the vectors between the points, computes a valid normal vector via the cross product, and determines the constant term using one of the given points. The final equation is correct.
gpt-oss:20b: pass 2026-09-29

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 2026-09-29 with SymPy 1.14.0.