Equation of a plane
Problem 9.41 · hard
Find an equation of the plane through the points \( \displaystyle (3, 3, -3),\ (-2, 2, -3),\ (-2, -3, -1) \).
- \[ \left[\begin{matrix}-5\\-1\\0\end{matrix}\right] \]PQ.Reviewed
- \[ \left[\begin{matrix}-5\\-6\\2\end{matrix}\right] \]PR.Reviewed
- \[ \left[\begin{matrix}-2\\10\\25\end{matrix}\right] \]A normal vector n = PQ × PR (scaled).Reviewed
- \[ -51 \]n · P gives the constant.Reviewed
- The plane is -2*x + 10*y + 25*z = -51.Reviewed
Answer \( - 2 x + 10 y + 25 z = -51 \)
✓ 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 one of the given points. 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 one of the given points. 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 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 matches the stated answer.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — 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: 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.