Equation of a plane
Problem 9.87 · medium
Find an equation of the plane through the points \( \displaystyle (3, -3, -3),\ (-3, 3, 1),\ (2, -2, 3) \).
- \[ \left[\begin{matrix}-6\\6\\4\end{matrix}\right] \]PQ.
- \[ \left[\begin{matrix}-1\\1\\6\end{matrix}\right] \]PR.
- \[ \left[\begin{matrix}1\\1\\0\end{matrix}\right] \]A normal vector n = PQ × PR (scaled).
- \[ 0 \]n · P gives the constant.
- The plane is x + y = 0.
Answer \( x + y = 0 \)
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: passqwen3.6:27b-mlx: fail (error) — The normal vector calculation is incorrect. The cross product of PQ = <-6, 6, 4> and PR = <-1, 1, 6> is <32, 32, 0>, not <1, 1, 0>. While the resulting plane equation x + y = 0 is coincidentally correct because the z-component of the normal is 0 and the constant term is 0, the intermediate vector arithmetic shown in step 3 is wrong (it appears to be a scaled version of the difference of the first two components only, ignoring the z-components entirely or miscalculating the cross product). A student following this logic would learn an incorrect method for finding normal vectors.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The normal vector calculation is incorrect. The cross product of PQ = <-6, 6, 4> and PR = <-1, 1, 6> is <32, 32, 0>, not <1, 1, 0>. While the resulting plane equation x + y = 0 is coincidentally correct because the z-component of the normal is 0 and the constant term is 0, the intermediate vector arithmetic shown in step 3 is wrong (it appears to be a scaled version of the difference of the first two components only, ignoring the z-components entirely or miscalculating the cross product). A student following this logic would learn an incorrect method for finding normal vectors.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-27 — The normal vector calculation is incorrect. The cross product of PQ = <-6, 6, 4> and PR = <-1, 1, 6> is <28, 32, 0>, not <1, 1, 0>. Consequently, the derived equation x + y = 0 is wrong (the correct equation is 7x + 8y = 3).gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — The normal vector calculation is incorrect. The cross product of PQ = <-6, 6, 4> and PR = <-1, 1, 6> is <28, 32, 0>, not <1, 1, 0>. Consequently, the derived equation x + y = 0 is wrong (the correct plane is 7x + 8y + 3 = 0).gpt-oss:20b: fail (error) 2026-09-26 — The cross product of PQ and PR is (32,40,0), not (1,1,0). Scaling to (1,1,0) is incorrect, so the normal vector is wrong, though the final plane equation happens to be correct.
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.