Home›Calculus 3›Equation of a plane›Problem 9.45

Equation of a plane

Problem 9.45 · hard

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

\[ \left[\begin{matrix}0\\-3\\2\end{matrix}\right] \]
PQ.Reviewed
\[ \left[\begin{matrix}-4\\3\\3\end{matrix}\right] \]
PR.Reviewed
\[ \left[\begin{matrix}-15\\-8\\-12\end{matrix}\right] \]
A normal vector n = PQ × PR (scaled).Reviewed
\[ -18 \]
n · P gives the constant.Reviewed
The plane is -15*x - 8*y - 12*z = -18.
Reviewed

Answer \( - 15 x - 8 y - 12 z = -18 \)

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 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 algebraically correct and 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 displacement vectors and determines the constant term using the dot product with a point on the plane. The final equation is algebraically correct and matches the stated answer.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the vectors, computes the normal vector via cross product, and determines the plane equation. The steps are logically sound and the final equation is correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies two vectors in the plane, computes their cross product to find a normal vector, 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) 2026-09-26 — The cross product PQ×PR was computed incorrectly: the y‑component should be +8, not –8. Consequently the normal vector and the plane equation are wrong.

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.