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Cross product

Problem 9.174 · easy

Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle -1, -4, -4 \rangle,\ \mathbf{v} = \langle -2, 1, -3 \rangle \), and the area of the parallelogram they span.

Expand the determinant with rows i j k, u, v.
Reviewed
\[ \left[\begin{matrix}16\\5\\-9\end{matrix}\right] \]
The components.✓ Proved
\[ \sqrt{362} \]
Area = |u × v|.✓ Proved

Answer \( \mathbf{u}\times\mathbf{v} = \langle 16, 5, -9 \rangle,\quad \text{area} = \sqrt{362} \)

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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	Lagrange's identity √(\|u\|²\|v\|² − (u·v)²) gives the same area

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the method for computing the cross product and the area of the parallelogram. The intermediate vector and final magnitude are correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the method for computing the cross product and the area of the parallelogram. The intermediate vector and final magnitude are correct.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The cross product calculation is incorrect. The correct cross product is <8, 11, 9>, not <16, 5, -9>. Consequently, the area is sqrt(226), not sqrt(362).
gpt-oss:20b: pass 2026-09-29

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/cross_product_area, checked 2026-09-29 with SymPy 1.14.0.