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Equation of a plane

Problem 9.108 · hard

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

\[ \left[\begin{matrix}2\\2\\1\end{matrix}\right] \]
PQ.✓ Proved
\[ \left[\begin{matrix}-1\\3\\-1\end{matrix}\right] \]
PR.✓ Proved
\[ \left[\begin{matrix}-5\\1\\8\end{matrix}\right] \]
A normal vector n = PQ × PR (scaled).✓ Proved
\[ -10 \]
n · P gives the constant.✓ Proved
The plane is -5*x + y + 8*z = -10.
Reviewed

Answer \( - 5 x + y + 8 z = -10 \)

✓ 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 computes the normal vector via the cross product of two vectors lying in the plane and determines the constant term using a point on the plane. The final equation is correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly computes the normal vector via the cross product of two vectors lying in the plane and determines the constant term using a point on the plane. The final equation is correct.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly computes the normal vector via the cross product of displacement vectors and determines the constant term using one of the given points. The final equation is correct.
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 2026-09-26 with SymPy 1.14.0.