Equation of a plane
Problem 9.159 · medium
Find an equation of the plane through the points \( \displaystyle (2, -2, 3),\ (-2, 2, 3),\ (-1, -3, 1) \).
- \[ \left[\begin{matrix}-4\\4\\0\end{matrix}\right] \]PQ.Reviewed
- \[ \left[\begin{matrix}-3\\-1\\-2\end{matrix}\right] \]PR.Reviewed
- \[ \left[\begin{matrix}-1\\-1\\2\end{matrix}\right] \]A normal vector n = PQ × PR (scaled).Reviewed
- \[ 6 \]n · P gives the constant.Reviewed
- The plane is -x - y + 2*z = 6.Reviewed
Answer \( - x - y + 2 z = 6 \)
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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the normal vector via the cross product of displacement vectors and verifies the plane equation using one of the given points. The final equation is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies the normal vector via the cross product of displacement vectors and verifies the plane equation using one of the given points. The final equation is correct.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the vectors, computes a valid normal vector, and determines the constant term. The final equation is correct.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-28 — The vector calculations for PQ and PR are incorrect. PQ should be <-4, 4, 0> and PR should be <-3, -1, -2>, but the cross product calculation in step 3 yields <-1, -1, 2> which is incorrect (the correct normal is <-8, -8, 16> or simplified <-1, -1, 2> is actually correct for the direction, wait. Let's recheck. PQ = <-4, 4, 0>. PR = <-3, -1, -2>. Cross product: i(4*-2 - 0*-1) - j(-4*-2 - 0*-3) + k(-4*-1 - 4*-3) = i(-8) - j(8) + k(4+12) = <-8, -8, 16>. Dividing by 8 gives <-1, -1, 2>. So the normal vector is correct. Let's check the constant. n . P = <-1, -1, 2> . <2, -2, 3> = -2 + 2 + 6 = 6. The equation -x -y + 2z = 6 is correct. Why did I think it was wrong? Let me re-read the solution steps carefully. Step 1 says PQ is <-4, 4, 0>. Step 2 says PR is <-3, -1, -2>. Step 3 says n is <-1, -1, 2>. This is a scaled version of the cross product. Step 4 says n . P is 6. Step 5 states the equation. The solution is actually correct. Wait, let me look at the 'unchecked' tags. The prompt says 'Each line ... is either an EQUATION ... or a SENTENCE'. Steps 1-4 are marked [equation, unchecked]. This implies the algebra was checked by a CAS? No, it says 'unchecked'. But the prompt says 'a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. These are marked 'unchecked'. This usually means I should check them? Or does it mean the CAS didn't check them? The prompt says 'do not re-check the algebra of a line marked proved'. These are NOT marked proved. They are marked unchecked. This suggests I SHOULD check them. Let's re-verify. P=(2,-2,3), Q=(-2,2,3), R=(-1,-3,1). PQ = Q-P = <-4, 4, 0>. Correct. PR = R-P = <-3, -1, -2>. Correct. PQ x PR = <-8, -8, 16>. The solution lists <-1, -1, 2>. This is a valid normal vector (scaled by 1/8). n . P = <-1, -1, 2> . <2, -2, 3> = -2 + 2 + 6 = 6. Correct. Equation: -x -y + 2z = 6. Correct. So why would it fail? Is there a subtle error? Maybe the points are collinear? No, vectors are not parallel. Maybe the question asks for a specific form? No. Maybe the 'unchecked' status implies I must verify the arithmetic, and I did, and it's correct. So it should pass. Let me double check the cross product again. i(4*-2 - 0) = -8. j(0 - (-4*-2)) = j(0-8) = -8j -> component is -8. k(-4*-1 - 4*-3) = 4 + 12 = 16. Vector <-8, -8, 16>. Simplified <-1, -1, 2>. Correct. Dot product with (2, -2, 3): -1(2) + -1(-2) + 2(3) = -2 + 2 + 6 = 6. Correct. The solution is sound. I will mark it pass.gpt-oss:20b: pass 2026-09-28
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.