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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.Reviewed
\[ 2 \sqrt{110} \]
Area = |u × v|.Reviewed

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

Lines: 3 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. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
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 the area of the parallelogram. The steps are logically sound and the final answer matches the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly computes the cross product and the area of the parallelogram. The steps are logically sound and the final answer matches the stated answer.
gpt-oss:20b: pass 2026-10-03
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 2026-10-13 with SymPy 1.14.0.