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Triple products and volumes

Problem 9.570 · easy

Find the volume of the parallelepiped determined by \( \displaystyle \left\langle 0, -2, 2 \right\rangle \), \( \displaystyle \left\langle 0, -3, -3 \right\rangle \) and \( \displaystyle \left\langle -4, 3, 4 \right\rangle \).

\[ \left[\begin{matrix}-3\\12\\-12\end{matrix}\right] \]
v × w.✓ Proved
\[ -48 \]
u·(v × w).✓ Proved
The volume is the absolute value of the triple product.
Reviewed

Answer \( 48 \)

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 computes the scalar triple product and applies the definition of volume as its absolute value. The final answer 48 is correct.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-13
qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly computes the scalar triple product and applies the definition of volume as its absolute value. The final answer 48 is correct.
qwen3.6:27b-mlx: pass 2026-10-13 — 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, so we assume they are correct, and the final logical step is sound.
gpt-oss:20b: fail (misleading) 2026-10-13 — The cross product of v and w is computed incorrectly (the j‑component should be –12, not +12). Although the final triple product happens to be correct, the intermediate step is false and could mislead a student.

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.