Equation of a plane
Problem 9.135 · hard
Find an equation of the plane through the points \( \displaystyle (0, 2, 0),\ (2, 3, 2),\ (-1, 2, -2) \).
- \[ \left[\begin{matrix}2\\1\\2\end{matrix}\right] \]PQ.✓ Proved
- \[ \left[\begin{matrix}-1\\0\\-2\end{matrix}\right] \]PR.✓ Proved
- \[ \left[\begin{matrix}-2\\2\\1\end{matrix}\right] \]A normal vector n = PQ × PR (scaled).✓ Proved
- \[ 4 \]n · P gives the constant.✓ Proved
- The plane is -2*x + 2*y + z = 4.Reviewed
Answer \( - 2 x + 2 y + z = 4 \)
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: passqwen3.6:27b-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 2026-09-27 — 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 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the normal vector via the cross product of two displacement vectors and determines the constant term using a point on the plane. The final equation is correct.gpt-oss:20b: fail (error) 2026-09-27 — The cross product of PQ and PR was computed incorrectly; the j‑component should be 0, not 2. Consequently the normal vector and the resulting plane equation are wrong.
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 2026-09-27 with SymPy 1.14.0.