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Cross product

Problem 9.149 · medium

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

Expand the determinant with rows i j k, u, v.
Reviewed
\[ \left[\begin{matrix}-10\\-14\\12\end{matrix}\right] \]
The components.✓ Proved
\[ 2 \sqrt{110} \]
Area = |u × v|.✓ Proved

Answer \( \mathbf{u}\times\mathbf{v} = \langle -10, -14, 12 \rangle,\quad \text{area} = 2 \sqrt{110} \)

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 computes the cross product and its magnitude. The steps are logically sound and the final answer matches the stated result.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly computes the cross product and its magnitude. The steps are logically sound and the final answer matches the stated result.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The cross product calculation is incorrect. The correct cross product is < -18, 24, 12 >, not < -10, -14, 12 >. Consequently, the area calculation is also wrong.
gpt-oss:20b: pass 2026-09-28

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 2026-09-28 with SymPy 1.14.0.