Home›Calculus 3›Triple products and volumes›Problem 9.408

Triple products and volumes

Problem 9.408 · easy

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

\[ \left[\begin{matrix}-8\\4\\4\end{matrix}\right] \]
v × w.✓ Proved
\[ -4 \]
u·(v × w).✓ Proved
The volume is the absolute value of the triple product.

Answer \( 4 \)

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 solution calculates the scalar triple product as -4 but fails to take the absolute value in the final answer, resulting in an incorrect volume of -4 instead of 4.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution calculates the scalar triple product as -4 but fails to take the absolute value in the final answer, resulting in an incorrect volume of -4 instead of 4.
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The cross product calculation in step 1 is incorrect; the first component should be -4, not -8. Consequently, the scalar triple product in step 2 is also incorrect (it should be 0, not -4), leading to a wrong final answer.
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.