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Triple products and volumes

Problem 9.485 · easy

Find the volume of the parallelepiped determined by \( \displaystyle \left\langle -4, 2, -2 \right\rangle \), \( \displaystyle \left\langle 4, -1, -2 \right\rangle \) and \( \displaystyle \left\langle 0, -2, 8 \right\rangle \).

\[ \left[\begin{matrix}-12\\-32\\-8\end{matrix}\right] \]
v × w.✓ Proved
\[ 0 \]
u·(v × w).✓ Proved
The volume is the absolute value of the triple product — 0 here, so the vectors are coplanar.

Answer \( 0 \)

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: pass
qwen3.6:27b-mlx: fail (error) — The cross product calculation in step 1 is incorrect. The cross product of <4, -1, -2> and <0, -2, 8> is <-4, -32, -8>, not <-12, -32, -8>. Consequently, the triple product is not zero, and the volume is not zero.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-11 — The cross product calculation in step 1 is incorrect. The cross product of <4, -1, -2> and <0, -2, 8> is <-4, -32, -8>, not <-12, -32, -8>. Consequently, the triple product is not zero, and the volume is not zero.
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The cross product calculation is incorrect; the actual cross product is <12, 32, 8>, not <-12, -32, -8>. Although the final volume of 0 is coincidentally correct because the vectors are indeed coplanar, the intermediate step contains a sign error that would mislead a student checking their work.
gpt-oss:20b: pass 2026-10-11

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 2026-10-11 with SymPy 1.14.0.