Home›Calculus 3›Equation of a plane›Problem 9.161

Equation of a plane

Problem 9.161 · hard

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

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

Answer \( - 2 y - z = -2 \)

Lines: 4 proved, 1 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.

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	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: pass
qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object

Every verdict on record (4)

qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer returned a non-object
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The vectors PQ and PR are calculated incorrectly. PQ should be (-1, 0, 0) but PR is calculated as (-2, 1, -2) instead of the correct (-1, 1, -2). This leads to an incorrect normal vector and final equation.
gpt-oss:20b: fail (error) 2026-09-28 — The cross product PQ×PR was computed with a wrong sign for the y‑component, leading to an incorrect normal vector. Consequently the constant term and the final plane equation are incorrect.

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 2026-09-28 with SymPy 1.14.0.