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.
- \[ \left[\begin{matrix}1\\-2\\0\end{matrix}\right] \]PR.
- \[ \left[\begin{matrix}-2\\-1\\-1\end{matrix}\right] \]A normal vector n = PQ × PR (scaled).
- \[ -3 \]n · P gives the constant.
- The plane is -2*x - y - z = -3.
Answer \( - 2 x - y - z = -3 \)
Lines: 5 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. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 2 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 3 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 4 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 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 (y‑component sign error), leading to an incorrect normal vector and constant. The resulting plane equation does not satisfy the given points.qwen3.6:27b-mlx: pass — The solution correctly identifies the normal vector via the cross product of displacement vectors and computes the plane equation. The final equation matches the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the normal vector via the cross product of displacement vectors and computes the plane equation. The final equation matches the stated answer.gpt-oss:20b: fail (error) 2026-09-28 — The cross product of PQ and PR was computed incorrectly (y‑component sign error), leading to an incorrect normal vector and constant. The resulting plane equation does not satisfy the given points.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 2026-10-13 with SymPy 1.14.0.