Equation of a plane
Problem 9.186 · hard
Find an equation of the plane through the points \( \displaystyle (-3, -1, -1),\ (2, -1, -3),\ (-2, -3, 3) \).
- \[ \left[\begin{matrix}5\\0\\-2\end{matrix}\right] \]PQ.✓ Proved
- \[ \left[\begin{matrix}1\\-2\\4\end{matrix}\right] \]PR.✓ Proved
- \[ \left[\begin{matrix}-2\\-11\\-5\end{matrix}\right] \]A normal vector n = PQ × PR (scaled).✓ Proved
- \[ 22 \]n · P gives the constant.✓ Proved
- The plane is -2*x - 11*y - 5*z = 22.Reviewed
Answer \( - 2 x - 11 y - 5 z = 22 \)
✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 two vectors lying on the plane, calculates 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 (4)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly computes two vectors lying on the plane, calculates 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 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — 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 2026-09-29
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 2026-09-29 with SymPy 1.14.0.