Triple products and volumes
Problem 9.481 · easy
Find the volume of the parallelepiped determined by \( \displaystyle \left\langle -1, -2, 0 \right\rangle \), \( \displaystyle \left\langle 2, 2, -2 \right\rangle \) and \( \displaystyle \left\langle -2, 0, 2 \right\rangle \).
- \[ \left[\begin{matrix}4\\0\\4\end{matrix}\right] \]v × w.✓ Proved
- \[ -4 \]u·(v × w).✓ Proved
- The volume is the absolute value of the triple product.Reviewed
Answer \( 4 \)
Lines: 2 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 | 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 | the 3 × 3 determinant, computed by mpmath |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to explicitly state the vectors u, v, and w used in the calculation, making it impossible to verify that the cross product and dot product correspond to the vectors given in the problem statement. Additionally, the intermediate steps are presented as unchecked equations without context, obscuring the logical flow.gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.