Home›Calculus 3›Equation of a plane›Problem 9.37

Equation of a plane

Problem 9.37 · hard

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

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

Answer \( 2 x + 3 y - 2 z = 1 \)

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: pass
qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer returned a non-object
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 computes the constant term accurately. The final equation matches the stated answer.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The vector calculations in steps 1 and 2 are incorrect. PQ should be (-1, 2, 2) but PR is calculated as (1, 2, 4) instead of the correct (1, 2, 4) -- wait, let's recheck. P=(-2,1,-1), R=(-1,3,3). PR = R-P = (1, 2, 4). So step 2 is actually correct. Let's re-read step 1. Q=(-3,3,1). PQ = Q-P = (-1, 2, 2). Step 1 is correct. Step 3: Cross product of (-1, 2, 2) and (1, 2, 4). i(8-4) - j(-4-2) + k(-2-2) = 4i + 6j - 4k. The solution claims (2, 3, -2). This is a scaled version (divided by 2). The comment says 'scaled', which is acceptable for finding a normal. Step 4: n dot P. Using n=(2,3,-2) and P=(-2,1,-1). 2(-2) + 3(1) - 2(-1) = -4 + 3 + 2 = 1. The constant is 1. The equation is 2x + 3y - 2z = 1. The solution is actually correct. Why did I think it was wrong? Let me double check the cross product again. (-1, 2, 2) x (1, 2, 4). x-comp: 2*4 - 2*2 = 8-4=4. y-comp: -( -1*4 - 2*1 ) = -( -4 - 2 ) = 6. z-comp: -1*2 - 2*1 = -2-2=-4. Vector is (4, 6, -4). Dividing by 2 gives (2, 3, -2). This is a valid normal vector. The constant calculation is correct. The final equation is correct. The solution is sound.
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.