Triple products and volumes
Problem 9.244 · easy
Find the volume of the parallelepiped determined by \( \displaystyle \left\langle 4, 1, -4 \right\rangle \), \( \displaystyle \left\langle -4, 2, -2 \right\rangle \) and \( \displaystyle \left\langle -4, -3, -3 \right\rangle \).
- \[ \left[\begin{matrix}-12\\-4\\20\end{matrix}\right] \]v × w.✓ Proved
- \[ -132 \]u·(v × w).✓ Proved
- The volume is the absolute value of the triple product.Reviewed
Answer \( 132 \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 — The solution correctly computes the scalar triple product and applies the absolute value to find the volume.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly computes the scalar triple product and applies the absolute value to find the volume.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the volume as the absolute value of the scalar triple product and the calculations are consistent with the stated answer.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.