Equation of a plane
Problem 9.184 · hard
Find an equation of the plane through the points \( \displaystyle (-1, 2, -1),\ (-3, 3, 2),\ (2, 1, -1) \).
- \[ \left[\begin{matrix}-2\\1\\3\end{matrix}\right] \]PQ.Reviewed
- \[ \left[\begin{matrix}3\\-1\\0\end{matrix}\right] \]PR.Reviewed
- \[ \left[\begin{matrix}3\\9\\-1\end{matrix}\right] \]A normal vector n = PQ × PR (scaled).Reviewed
- \[ 16 \]n · P gives the constant.Reviewed
- The plane is 3*x + 9*y - z = 16.Reviewed
Answer \( 3 x + 9 y - z = 16 \)
✓ 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 verifies the constant term. The final equation is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly computes the normal vector via the cross product of displacement vectors and verifies the constant term. The final equation is correct.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-09-29 — 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 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the vectors between the points, computes a valid normal vector via the cross product, and determines the constant term using one of the given points. The final equation is correct.gpt-oss:20b: pass 2026-09-29
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.