Equation of a plane
Problem 9.134 · hard
Find an equation of the plane through the points \( \displaystyle (1, -1, 1),\ (-2, 2, -3),\ (-2, 3, 2) \).
- \[ \left[\begin{matrix}-3\\3\\-4\end{matrix}\right] \]PQ.
- \[ \left[\begin{matrix}-3\\4\\1\end{matrix}\right] \]PR.
- \[ \left[\begin{matrix}19\\15\\-3\end{matrix}\right] \]A normal vector n = PQ × PR (scaled).
- \[ 1 \]n · P gives the constant.
- The plane is 19*x + 15*y - 3*z = 1.
Answer \( 19 x + 15 y - 3 z = 1 \)
Lines: 5 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 2 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 3 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 4 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 5 | Not checked | — | 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: fail (error) — The cross product PQ×PR yields (19, –15, –3), not (19, 15, –3). Consequently the normal vector and the constant term are wrong, so the plane equation 19*x+15*y–3*z=1 does not pass through the given points.qwen3.6:27b-mlx: pass — The solution correctly computes the normal vector via the cross product of two vectors lying in the plane and determines the constant term using a point on the plane. The final equation matches the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly computes the normal vector via the cross product of two vectors lying in the plane and determines the constant term using a point on the plane. The final equation matches the stated answer.gpt-oss:20b: fail (error) 2026-09-28 — The cross product PQ×PR yields (19, –15, –3), not (19, 15, –3). Consequently the normal vector and the constant term are wrong, so the plane equation 19*x+15*y–3*z=1 does not pass through the given points.qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the normal vector via the cross product of two displacement vectors and determines the constant term using the dot product with a point on the plane. The final equation is correct.gpt-oss:20b: fail (error) 2026-09-27 — The cross product of PQ and PR is (19,-15,-3), not (19,15,-3). Consequently the normal vector and the constant term are incorrect, leading to an incorrect plane equation.qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the normal vector via the cross product of displacement vectors and determines the constant term using one of the given points. The final equation is correct.gpt-oss:20b: fail (error) 2026-09-27 — The cross product PQ×PR is (19,-15,-3), not (19,15,-3). Consequently the normal vector is wrong and the resulting plane equation 19x+15y-3z=1 does not pass through the given points.
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.