Home›Calculus 3›Equation of a plane›Problem 9.185

Equation of a plane

Problem 9.185 · hard

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

\[ \left[\begin{matrix}-5\\1\\-4\end{matrix}\right] \]
PQ.Reviewed
\[ \left[\begin{matrix}-5\\-2\\-2\end{matrix}\right] \]
PR.Reviewed
\[ \left[\begin{matrix}-2\\2\\3\end{matrix}\right] \]
A normal vector n = PQ × PR (scaled).Reviewed
\[ 0 \]
n · P gives the constant.Reviewed
The plane is -2*x + 2*y + 3*z = 0.
Reviewed

Answer \( - 2 x + 2 y + 3 z = 0 \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the normal vector via the cross product of displacement vectors and verifies the constant term using one of the given points. The final equation is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the normal vector via the cross product of displacement vectors and verifies the constant term using one of the given points. The final equation is correct.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The normal vector calculation is incorrect. The cross product of PQ = (-5, 1, -4) and PR = (-5, -2, -2) is (-10, 30, 15), which simplifies to (-2, 6, 3), not (-2, 2, 3). Consequently, the derived plane equation is wrong.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The normal vector calculation is incorrect. The cross product of vectors PQ and PR does not yield (-2, 2, 3); for example, the z-component should be (-5)(-2) - (1)(-5) = 15, not 3. Consequently, the derived plane equation is wrong.
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 2026-10-13 with SymPy 1.14.0.