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Equation of a plane

Problem 9.205 · hard

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

\[ \left[\begin{matrix}-5\\-2\\3\end{matrix}\right] \]
PQ.✓ Proved
\[ \left[\begin{matrix}0\\-3\\2\end{matrix}\right] \]
PR.✓ Proved
\[ \left[\begin{matrix}1\\2\\3\end{matrix}\right] \]
A normal vector n = PQ × PR (scaled).✓ Proved
\[ -4 \]
n · P gives the constant.✓ Proved
The plane is x + 2*y + 3*z = -4.
Reviewed

Answer \( x + 2 y + 3 z = -4 \)

✓ 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 the normal vector via the cross product of displacement vectors and verifies the constant term. The final equation is correct.

Every verdict on record (4)

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. The final equation is correct.
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 computes the constant term accurately. The final equation matches the stated answer.
gpt-oss:20b: pass 2026-10-03

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.