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Equation of a plane

Problem 9.111 · medium

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

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

Answer \( 5 x + y = 3 \)

Lines: 4 proved, 1 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.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
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: fail (error) — The vector calculations in steps 1 and 2 are incorrect. Given points P(1, -2, -1), Q(1, -2, -2), R(0, 3, 1), the vector PQ should be <0, 0, -1> (correct in step 1) but PR should be <-1, 5, 2> (correct in step 2). However, the cross product PQ x PR = <0, 0, -1> x <-1, 5, 2> = <(-1)(5) - (0)(2), (0)(-1) - (-1)(-1), (0)(5) - (0)(-1)> = <-5, -1, 0>. The solution claims the normal is <5, 1, 0>, which is parallel to <-5, -1, 0>, so the normal direction is actually correct (just scaled by -1). Let's re-read carefully. Step 1 says PQ is <0,0,-1>. Step 2 says PR is <-1, 5, 2>. Step 3 says n = <5, 1, 0>. Let's check the cross product again. i(0*2 - (-1)*5) - j(0*2 - (-1)*(-1)) + k(0*5 - 0*(-1)) = i(5) - j(-1) + k(0) = <5, 1, 0>. Okay, the normal vector calculation is actually correct. Let's check the plane equation. Normal <5, 1, 0>. Point P(1, -2, -1). 5(1) + 1(-2) + 0(-1) = 5 - 2 = 3. So 5x + y = 3. Check other points. Q(1, -2, -2): 5(1) + (-2) = 3. Correct. R(0, 3, 1): 5(0) + 3 = 3. Correct. The solution is mathematically sound. Why did I think it was wrong? I misread the cross product sign initially. The steps are correct. The sentence in step 5 correctly states the final equation derived from the previous steps.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-26 — The vector calculations in steps 1 and 2 are incorrect. Given points P(1, -2, -1), Q(1, -2, -2), R(0, 3, 1), the vector PQ should be <0, 0, -1> (correct in step 1) but PR should be <-1, 5, 2> (correct in step 2). However, the cross product PQ x PR = <0, 0, -1> x <-1, 5, 2> = <(-1)(5) - (0)(2), (0)(-1) - (-1)(-1), (0)(5) - (0)(-1)> = <-5, -1, 0>. The solution claims the normal is <5, 1, 0>, which is parallel to <-5, -1, 0>, so the normal direction is actually correct (just scaled by -1). Let's re-read carefully. Step 1 says PQ is <0,0,-1>. Step 2 says PR is <-1, 5, 2>. Step 3 says n = <5, 1, 0>. Let's check the cross product again. i(0*2 - (-1)*5) - j(0*2 - (-1)*(-1)) + k(0*5 - 0*(-1)) = i(5) - j(-1) + k(0) = <5, 1, 0>. Okay, the normal vector calculation is actually correct. Let's check the plane equation. Normal <5, 1, 0>. Point P(1, -2, -1). 5(1) + 1(-2) + 0(-1) = 5 - 2 = 3. So 5x + y = 3. Check other points. Q(1, -2, -2): 5(1) + (-2) = 3. Correct. R(0, 3, 1): 5(0) + 3 = 3. Correct. The solution is mathematically sound. Why did I think it was wrong? I misread the cross product sign initially. The steps are correct. The sentence in step 5 correctly states the final equation derived from the previous steps.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer returned a non-object
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 2026-09-26 with SymPy 1.14.0.