Equation of a plane
Problem 9.80 · hard
Find an equation of the plane through the points \( \displaystyle (-2, -1, -2),\ (-3, 2, 2),\ (1, -1, 1) \).
- \[ \left[\begin{matrix}-1\\3\\4\end{matrix}\right] \]PQ.Reviewed
- \[ \left[\begin{matrix}3\\0\\3\end{matrix}\right] \]PR.Reviewed
- \[ \left[\begin{matrix}3\\5\\-3\end{matrix}\right] \]A normal vector n = PQ × PR (scaled).Reviewed
- \[ -5 \]n · P gives the constant.Reviewed
- The plane is 3*x + 5*y - 3*z = -5.Reviewed
Answer \( 3 x + 5 y - 3 z = -5 \)
Lines: 5 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. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 3 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 4 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 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 computes the normal vector via the cross product of displacement vectors and verifies the constant term. The final equation matches the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly computes the normal vector via the cross product of displacement vectors and verifies the constant term. The final equation matches the stated answer.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the normal vector via the cross product of displacement vectors and determines the constant term using a point on the plane. The final equation matches the stated answer.gpt-oss:20b: fail (error) 2026-09-27 — The cross product PQ×PR is (9,-15,-9), not (3,5,-3). The normal vector’s y‑component has the wrong sign, leading to an incorrect plane equation.qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly computes the normal vector via the cross product of two displacement vectors and determines the constant term using one of the given points. The final equation is correct.gpt-oss:20b: pass 2026-09-26
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.