Equation of a plane
Problem 9.157 · hard
Find an equation of the plane through the points \( \displaystyle (-1, -1, 0),\ (1, 3, -1),\ (-3, 2, 3) \).
- \[ \left[\begin{matrix}2\\4\\-1\end{matrix}\right] \]PQ.Reviewed
- \[ \left[\begin{matrix}-2\\3\\3\end{matrix}\right] \]PR.Reviewed
- \[ \left[\begin{matrix}15\\-4\\14\end{matrix}\right] \]A normal vector n = PQ × PR (scaled).Reviewed
- \[ -11 \]n · P gives the constant.Reviewed
- The plane is 15*x - 4*y + 14*z = -11.Reviewed
Answer \( 15 x - 4 y + 14 z = -11 \)
✓ 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 two displacement vectors 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 2026-10-03 — 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 2026-10-03qwen3.6:27b-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 2026-09-28 — The solution correctly computes the normal vector via the cross product of two vectors lying on the plane and determines the constant term using one of the given points. The final equation is correct.gpt-oss:20b: pass 2026-09-28
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.