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Triple products and volumes

Problem 9.249 · easy

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

\[ \left[\begin{matrix}6\\5\\-4\end{matrix}\right] \]
v × w.✓ Proved
\[ 6 \]
u·(v × w).✓ Proved
The volume is the absolute value of the triple product.
Reviewed

Answer \( 6 \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly computes the cross product and dot product, and applies the standard formula for the volume of a parallelepiped.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly computes the cross product and dot product, and applies the standard formula for the volume of a parallelepiped.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the volume as the absolute value of the scalar triple product. Although the intermediate steps are presented as unchecked equations, the final sentence accurately states the geometric principle required to obtain the magnitude from the signed volume.
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.