Triple products and volumes
Problem 9.328 · easy
Find the volume of the parallelepiped determined by \( \displaystyle \left\langle 2, -4, -3 \right\rangle \), \( \displaystyle \left\langle -1, 4, -2 \right\rangle \) and \( \displaystyle \left\langle -3, -4, 3 \right\rangle \).
- \[ \left[\begin{matrix}4\\9\\16\end{matrix}\right] \]v × w.✓ Proved
- \[ -76 \]u·(v × w).✓ Proved
- The volume is the absolute value of the triple product.
Answer \( 76 \)
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 calculates the scalar triple product as -76 but fails to explicitly state the final volume as 76, leaving the answer implicit. Additionally, the intermediate cross product shown is incorrect for the given vectors, suggesting a calculation error in the setup even if the final scalar product magnitude happened to match the stated answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution calculates the scalar triple product as -76 but fails to explicitly state the final volume as 76, leaving the answer implicit. Additionally, the intermediate cross product shown is incorrect for the given vectors, suggesting a calculation error in the setup even if the final scalar product magnitude happened to match the stated answer.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The intermediate vector shown in step 1 is incorrect; the cross product of the given vectors is not <4, 9, 16>. Although the final scalar result (-76) happens to be correct, the displayed work contains a factual error that would mislead a student checking the intermediate steps.gpt-oss:20b: pass 2026-10-07
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 2026-10-07 with SymPy 1.14.0.