Triple products and volumes
Problem 9.246 · easy
Find the volume of the parallelepiped determined by \( \displaystyle \left\langle 0, -3, 3 \right\rangle \), \( \displaystyle \left\langle 1, 3, 3 \right\rangle \) and \( \displaystyle \left\langle 3, -1, 0 \right\rangle \).
- \[ \left[\begin{matrix}3\\9\\-10\end{matrix}\right] \]v × w.✓ Proved
- \[ -57 \]u·(v × w).✓ Proved
- The volume is the absolute value of the triple product.Reviewed
Answer \( 57 \)
✓ 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 definition that volume is the absolute value of this product.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly computes the scalar triple product and applies the definition that volume is the absolute value of this product.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. The intermediate calculations (cross product and dot product) are marked as unchecked equations, and the final result 57 is correct.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.