Equation of a plane
Problem 9.132 · hard
Find an equation of the plane through the points \( \displaystyle (1, -1, 2),\ (-1, -2, -2),\ (2, -3, 2) \).
- \[ \left[\begin{matrix}-2\\-1\\-4\end{matrix}\right] \]PQ.Reviewed
- \[ \left[\begin{matrix}1\\-2\\0\end{matrix}\right] \]PR.Reviewed
- \[ \left[\begin{matrix}-8\\-4\\5\end{matrix}\right] \]A normal vector n = PQ × PR (scaled).Reviewed
- \[ 6 \]n · P gives the constant.Reviewed
- The plane is -8*x - 4*y + 5*z = 6.Reviewed
Answer \( - 8 x - 4 y + 5 z = 6 \)
✓ 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 identifies two vectors lying on the plane, computes their cross product to find a normal vector, and determines the constant term using the dot product with a point on the plane. The final equation is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — 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 the dot product with a point on the plane. The final equation is correct.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — 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 the dot product with one of the points. The final equation is correct.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the vectors PQ and PR, their cross product to find the normal vector, and the constant term using the dot product. 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.