Home›Calculus 3›Equation of a plane›Problem 9.84

Equation of a plane

Problem 9.84 · hard

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

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

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

✓ 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 cross product and computes the plane equation. The intermediate steps are algebraically verified.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the normal vector via cross product and computes the plane equation. The intermediate steps are algebraically verified.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the normal vector via the cross product of displacement vectors and verifies the constant term. The final equation matches the stated answer.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the normal vector via cross product and verifies the constant term. The final equation matches the stated answer.
gpt-oss:20b: pass 2026-09-26

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.