Home›Calculus 3›Equation of a plane›Problem 9.42

Equation of a plane

Problem 9.42 · hard

Find an equation of the plane through the points \( \displaystyle (-2, 0, 3),\ (-3, 3, -3),\ (1, 3, -3) \).

\[ \left[\begin{matrix}-1\\3\\-6\end{matrix}\right] \]
PQ.Reviewed
\[ \left[\begin{matrix}3\\3\\-6\end{matrix}\right] \]
PR.Reviewed
\[ \left[\begin{matrix}0\\-2\\-1\end{matrix}\right] \]
A normal vector n = PQ × PR (scaled).Reviewed
\[ -3 \]
n · P gives the constant.Reviewed
The plane is -2*y - z = -3.
Reviewed

Answer \( - 2 y - z = -3 \)

Lines: 5 reviewed. 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	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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the normal vector and the constant term, resulting in the correct plane equation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the normal vector and the constant term, resulting in the correct plane equation.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the normal vector and the plane equation. The final equation -2y - z = -3 is satisfied by all three given points.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly computes the normal vector via cross product and determines the plane equation. The final equation is algebraically equivalent to the standard form derived from the points.
gpt-oss:20b: fail (misleading) 2026-09-26 — The normal vector’s sign was flipped, leading to the incorrect plane equation -2*y - z = -3. The correct equation is 2*y - z = -3.

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.