Triple products and volumes
Problem 9.404 · easy
Find the volume of the parallelepiped determined by \( \displaystyle \left\langle 4, -3, -4 \right\rangle \), \( \displaystyle \left\langle -3, 0, -3 \right\rangle \) and \( \displaystyle \left\langle -2, -3, 3 \right\rangle \).
- \[ \left[\begin{matrix}-9\\15\\9\end{matrix}\right] \]v × w.✓ Proved
- \[ -117 \]u·(v × w).✓ Proved
- The volume is the absolute value of the triple product.
Answer \( 117 \)
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: fail (error) — The cross product v×w is computed incorrectly (the second component should be –15, not +15). Although the final triple product happens to equal –117, the intermediate step is algebraically wrong and could mislead a student.qwen3.6:27b-mlx: pass — The solution correctly identifies the volume as the absolute value of the scalar triple product and computes it accurately.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-08 — The cross product v×w is computed incorrectly (the second component should be –15, not +15). Although the final triple product happens to equal –117, the intermediate step is algebraically wrong and could mislead a student.qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the volume as the absolute value of the scalar triple product and computes it accurately.qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly computes the cross product and dot product, and correctly applies the definition of volume as the absolute value of the scalar triple product.gpt-oss:20b: pass 2026-10-08
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 2026-10-08 with SymPy 1.14.0.