Home›Calculus 3›Equation of a plane›Problem 9.88

Equation of a plane

Problem 9.88 · medium

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

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

Answer \( 2 x - z = 0 \)

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: fail (style) 2026-09-27 — [domain objection, downgraded to style] The vectors PQ and PR are calculated incorrectly. Given points P(0,1,0), Q(-1,3,-2), R(0,2,0), the vector PQ should be <-1, 2, -2> (correct in line 1) but PR should be <0, 1, 0> (correct in line 2). However, the cross product in line 3 is claimed to be <2, 0, -1>. Let's check: PQ x PR = <-1, 2, -2> x <0, 1, 0> = (2*0 - (-2)*1, -1*0 - (-2)*0, -1*1 - 2*0) = <2, 0, -1>. The cross product calculation is actually correct. Wait, let me re-read the points. P=(0,1,0), Q=(-1,3,-2), R=(0,2,0). PQ = Q-P = <-1, 2, -2>. PR = R-P = <0, 1, 0>. Cross product PQ x PR = <2, 0, -1>. The normal vector is correct. The equation 2x - z = 0 passes through P(0,1,0) -> 0-0=0. Q(-1,3,-2) -> 2(-1) - (-2) = -2+2=0. R(0,2,0) -> 0-0=0. The solution is actually correct. Why did I think it was wrong? Let me re-read the prompt's constraints. "Find what is wrong". Is there a subtle error? Line 1 says "PQ". Line 2 says "PR". Line 3 says "n = PQ x PR (scaled)". The result <2, 0, -1> is not scaled, it is the exact cross product. Calling it "scaled" is slightly misleading but not an error. However, look at line 4: "n . P gives the constant". n . P = <2,0,-1> . <0,1,0> = 0. The equation is 2x + 0y - z = 0. This is correct. Is there any issue? The problem asks for "an equation". The answer is correct. Let me look closer at the vectors. P=(0,1,0). Q=(-1,3,-2). R=(0,2,0). PQ = <-1, 2, -2>. PR = <0, 1, 0>. Cross product: i(2*0 - (-2)*1) - j((-1)*0 - (-2)*0) + k((-1)*1 - 2*0) = i(2) - j(0) + k(-1) = <2, 0, -1>. Everything seems correct. Wait, did I miss something? Ah, look at line 1 and 2. They are marked [equation, proved]. The text says "Matrix([...]), Matrix([...]) -- PQ." This is just defining the vectors. Line 3 defines the normal. Line 4 defines the constant. Line 5 states the equation. It seems correct. Let me double check the "scaled" comment. The cross product IS the normal. It is not scaled. Saying it is scaled implies it might not be the direct cross product, but it is. This is a minor style/misleading issue. But is it an error? No. Is there a bigger error? What if the points are collinear? PQ = <-1, 2, -2>. PR = <0, 1, 0>. They are not parallel. So a unique plane exists. The solution is sound. Why would it fail? Maybe the order of points? The problem lists (0,1,0), (-1,3,-2), (0,2,0). The solution uses P, Q, R. It doesn't explicitly define which is which, but the vectors match the standard assignment P=first, Q=second, R=third. This is standard convention. I will mark it as pass.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The vector labeled PQ is actually QR (or -PQ), and the vector labeled PR is actually OR (or -PR), not vectors originating from P. Although the cross product of any two non-parallel vectors in the plane yields a valid normal, the labels are factually incorrect and misleading regarding the geometric construction.
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.