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.✓ Proved
- \[ \left[\begin{matrix}-3\\-1\\-2\end{matrix}\right] \]PR.✓ Proved
- \[ \left[\begin{matrix}-1\\-1\\2\end{matrix}\right] \]A normal vector n = PQ × PR (scaled).✓ Proved
- \[ 6 \]n · P gives the constant.✓ Proved
- The plane is -x - y + 2*z = 6.Reviewed
Answer \( - x - y + 2 z = 6 \)
Lines: 4 proved, 1 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.
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 | 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 vectors, computes a valid normal vector, and determines the constant term. The final equation is correct.
Every verdict on record (4)
qwen3.6:27b-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 2026-09-28 with SymPy 1.14.0.