Equation of a plane
Problem 9.83 · hard
Find an equation of the plane through the points \( \displaystyle (-2, 2, -3),\ (0, 2, -2),\ (-2, -1, 0) \).
- \[ \left[\begin{matrix}2\\0\\1\end{matrix}\right] \]PQ.
- \[ \left[\begin{matrix}0\\-3\\3\end{matrix}\right] \]PR.
- \[ \left[\begin{matrix}1\\-2\\-2\end{matrix}\right] \]A normal vector n = PQ × PR (scaled).
- \[ 0 \]n · P gives the constant.
- The plane is x - 2*y - 2*z = 0.
Answer \( x - 2 y - 2 z = 0 \)
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: passqwen3.6:27b-mlx: fail (error) — The solution computes vectors PQ and PR incorrectly based on the given points. For P(-2, 2, -3), Q(0, 2, -2), and R(-2, -1, 0), PQ should be <2, 0, 1> and PR should be <0, -3, 3>. The solution lists PQ as <2, 0, 1> (correct) but PR as <0, -3, 3> (correct). However, the cross product calculation in step 3 yields <1, -2, -2>, which is incorrect. The actual cross product of <2, 0, 1> and <0, -3, 3> is <3, -6, -6>, which simplifies to <1, -2, -2>. Wait, let me re-check. 2*3 - 1*(-3) = 6+3=9? No. i(0*3 - 1*(-3)) - j(2*3 - 1*0) + k(2*(-3) - 0*0) = i(3) - j(6) + k(-6) = <3, -6, -6>. This simplifies to <1, -2, -2>. So the normal vector is correct. Let's check the plane equation. Using P(-2, 2, -3): 1(-2) - 2(2) - 2(-3) = -2 - 4 + 6 = 0. The equation x - 2y - 2z = 0 is correct for the points. Why did I think it was wrong? Let me re-read the vectors in the solution. Step 1 says PQ is <2, 0, 1>. Step 2 says PR is <0, -3, 3>. Step 3 says n is <1, -2, -2>. This is consistent. Step 4 says n.P gives 0. Step 5 gives the equation. The solution is actually correct. Let me double check the points again. P(-2, 2, -3), Q(0, 2, -2), R(-2, -1, 0). PQ = Q-P = <2, 0, 1>. PR = R-P = <0, -3, 3>. Cross product: i(0*3 - 1*(-3)) = 3i. j(2*3 - 1*0) = 6j -> -6j. k(2*(-3) - 0*0) = -6k. Vector <3, -6, -6>. Divide by 3: <1, -2, -2>. Equation: 1(x - (-2)) - 2(y - 2) - 2(z - (-3)) = 0 => x + 2 - 2y + 4 - 2z - 6 = 0 => x - 2y - 2z = 0. The solution is correct. I will mark it as pass.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution computes vectors PQ and PR incorrectly based on the given points. For P(-2, 2, -3), Q(0, 2, -2), and R(-2, -1, 0), PQ should be <2, 0, 1> and PR should be <0, -3, 3>. The solution lists PQ as <2, 0, 1> (correct) but PR as <0, -3, 3> (correct). However, the cross product calculation in step 3 yields <1, -2, -2>, which is incorrect. The actual cross product of <2, 0, 1> and <0, -3, 3> is <3, -6, -6>, which simplifies to <1, -2, -2>. Wait, let me re-check. 2*3 - 1*(-3) = 6+3=9? No. i(0*3 - 1*(-3)) - j(2*3 - 1*0) + k(2*(-3) - 0*0) = i(3) - j(6) + k(-6) = <3, -6, -6>. This simplifies to <1, -2, -2>. So the normal vector is correct. Let's check the plane equation. Using P(-2, 2, -3): 1(-2) - 2(2) - 2(-3) = -2 - 4 + 6 = 0. The equation x - 2y - 2z = 0 is correct for the points. Why did I think it was wrong? Let me re-read the vectors in the solution. Step 1 says PQ is <2, 0, 1>. Step 2 says PR is <0, -3, 3>. Step 3 says n is <1, -2, -2>. This is consistent. Step 4 says n.P gives 0. Step 5 gives the equation. The solution is actually correct. Let me double check the points again. P(-2, 2, -3), Q(0, 2, -2), R(-2, -1, 0). PQ = Q-P = <2, 0, 1>. PR = R-P = <0, -3, 3>. Cross product: i(0*3 - 1*(-3)) = 3i. j(2*3 - 1*0) = 6j -> -6j. k(2*(-3) - 0*0) = -6k. Vector <3, -6, -6>. Divide by 3: <1, -2, -2>. Equation: 1(x - (-2)) - 2(y - 2) - 2(z - (-3)) = 0 => x + 2 - 2y + 4 - 2z - 6 = 0 => x - 2y - 2z = 0. The solution is correct. I will mark it as pass.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: inconclusive 2026-09-27 — reviewer returned a non-objectgpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer returned a non-objectgpt-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.