Triple products and volumes
Problem 9.569 · easy
Are \( \displaystyle \mathbf u = \left\langle 4, -4, 1 \right\rangle \), \( \displaystyle \mathbf v = \left\langle 4, 0, 0 \right\rangle \) and \( \displaystyle \mathbf w = \left\langle 1, -2, 1 \right\rangle \) coplanar?
- \[ \left[\begin{matrix}0\\-4\\-8\end{matrix}\right] \]v × w.✓ Proved
- \[ 8 \]u·(v × w).✓ Proved
- Three vectors are coplanar exactly when u·(v × w) = 0.
Answer \( \text{not coplanar} \)
Lines: 2 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 | Not checked | — | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the 3 × 3 determinant, computed by mpmath |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution correctly calculates the scalar triple product as 8, but fails to state the conclusion that the vectors are not coplanar because the product is non-zero. It only states the condition for coplanarity without applying it to the result.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution correctly calculates the scalar triple product as 8, but fails to state the conclusion that the vectors are not coplanar because the product is non-zero. It only states the condition for coplanarity without applying it to the result.qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution calculates the scalar triple product as 8, which is non-zero, implying the vectors are not coplanar. However, the stated answer is 'not coplanar', which is consistent with the calculation. Wait, let me re-read the prompt's instruction: 'Stated answer: not coplanar'. The calculation shows u . (v x w) = 8 != 0. Therefore, they are NOT coplanar. The solution's logic is correct. Why would it fail? Let me check the cross product calculation. v = <4, 0, 0>, w = <1, -2, 1>. v x w = <0*1 - 0*(-2), 0*1 - 4*1, 4*(-2) - 0*1> = <0, -4, -8>. Correct. u = <4, -4, 1>. u . <0, -4, -8> = 4*0 + (-4)*(-4) + 1*(-8) = 0 + 16 - 8 = 8. Correct. The vectors are not coplanar. The stated answer is 'not coplanar'. The solution concludes they are not coplanar (implicitly, by showing the product is non-zero and stating the condition for coplanarity is 0). Is there a missing step? The solution lists the equation 8=8 and the sentence 'Three vectors are coplanar exactly when u·(v × w) = 0'. It does not explicitly state the conclusion 'Therefore, they are not coplanar'. However, usually, if the final answer matches the derivation, it's fine. But wait, looking at the provided solution text again: it stops at the theorem statement. It does not explicitly conclude 'Since 8 != 0, they are not coplanar'. Is this an error? Or just incomplete? The prompt asks to review the solution. If the solution doesn't explicitly draw the conclusion, it might be considered incomplete or misleading if a student doesn't make the final leap. But typically, 'pass' is for sound solutions. Let's look closer. Is there a subtle error? No. Is it misleading? A student might think the solution is just listing facts. But the 'Stated answer' is separate. The solution block itself ends with the theorem. It fails to apply the theorem to the calculated value. This is a logical gap. It's not an 'error' in calculation, but a failure to complete the argument. I will mark it as 'error' because the solution does not actually answer the question posed in the problem statement within the solution steps; it sets up the test but doesn't execute the final logical step.gpt-oss:20b: pass 2026-10-13
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/triple_product, checked 2026-10-13 with SymPy 1.14.0.