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Triple products and volumes

Problem 9.564 · easy

Find the volume of the parallelepiped determined by \( \displaystyle \left\langle 3, 2, -4 \right\rangle \), \( \displaystyle \left\langle -4, -1, -1 \right\rangle \) and \( \displaystyle \left\langle -4, 2, 3 \right\rangle \).

\[ \left[\begin{matrix}-1\\16\\-12\end{matrix}\right] \]
v × w.✓ Proved
\[ 77 \]
u·(v × w).✓ Proved
The volume is the absolute value of the triple product.
Reviewed

Answer \( 77 \)

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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the volume as the absolute value of the scalar triple product and the calculations are verified.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-13
qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the volume as the absolute value of the scalar triple product and the calculations are verified.
gpt-oss:20b: fail (error) 2026-10-13 — The cross product v×w was computed incorrectly (the j‑component sign is wrong). Consequently the triple product u·(v×w) is 13, not 77, so the stated volume is incorrect.
qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the volume as the absolute value of the scalar triple product and the numerical result is correct.

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 2026-10-13 with SymPy 1.14.0.