Equation of a plane
Problem 9.155 · hard
Find an equation of the plane through the points \( \displaystyle (3, 2, -2),\ (2, 2, 1),\ (2, 3, -3) \).
- \[ \left[\begin{matrix}-1\\0\\3\end{matrix}\right] \]PQ.Reviewed
- \[ \left[\begin{matrix}-1\\1\\-1\end{matrix}\right] \]PR.Reviewed
- \[ \left[\begin{matrix}-3\\-4\\-1\end{matrix}\right] \]A normal vector n = PQ × PR (scaled).Reviewed
- \[ -15 \]n · P gives the constant.Reviewed
- The plane is -3*x - 4*y - z = -15.Reviewed
Answer \( - 3 x - 4 y - z = -15 \)
✓ Nihil obstat Lines: 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 determines the constant term. The final equation is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly computes the normal vector via the cross product of displacement vectors and determines the constant term. The final equation is correct.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies two vectors in 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: fail (error) 2026-09-28 — The cross product of PQ and PR is (-3, 4, -1), not (-3, -4, -1). Using the correct normal vector gives the plane equation -3x + 4y - z = 1, not -3x - 4y - z = -15. The solution’s final equation is therefore incorrect.qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies two vectors in 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: fail (error) 2026-09-28 — The cross product of PQ and PR was computed incorrectly: the j‑component should be +4, not –4. Consequently the normal vector and the constant term are wrong, leading to an incorrect plane equation.
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.