Equation of a plane
Problem 9.137 · hard
Find an equation of the plane through the points \( \displaystyle (2, 1, -2),\ (3, 0, -3),\ (3, -1, -2) \).
- \[ \left[\begin{matrix}1\\-1\\-1\end{matrix}\right] \]PQ.✓ Proved
- \[ \left[\begin{matrix}1\\-2\\0\end{matrix}\right] \]PR.✓ Proved
- \[ \left[\begin{matrix}-2\\-1\\-1\end{matrix}\right] \]A normal vector n = PQ × PR (scaled).✓ Proved
- \[ -3 \]n · P gives the constant.✓ Proved
- The plane is -2*x - y - z = -3.
Answer \( - 2 x - y - z = -3 \)
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: fail (error) — The cross product of PQ and PR was computed incorrectly: the j‑component should be +1, not –1. Consequently the normal vector and the constant term are wrong, leading to an incorrect plane equation.qwen3.6:27b-mlx: pass — The solution correctly computes two vectors in the plane, finds a normal vector via cross product, and determines the constant term. The final equation is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes two vectors in the plane, finds a normal vector via cross product, and determines the constant term. 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 +1, not –1. Consequently the normal vector and the constant term are wrong, leading to an incorrect plane equation.qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies two vectors in the plane, computes a valid normal vector via cross product, and determines the constant term using one of the given points. The final equation is correct.gpt-oss:20b: pass 2026-09-27
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.