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Cross product

Problem 9.123 · medium

Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle 4, 0, 1 \rangle,\ \mathbf{v} = \langle -2, 2, -2 \rangle \), and the area of the parallelogram they span.

Expand the determinant with rows i j k, u, v.
Reviewed
\[ \left[\begin{matrix}-2\\6\\8\end{matrix}\right] \]
The components.✓ Proved
\[ 2 \sqrt{26} \]
Area = |u × v|.✓ Proved

Answer \( \mathbf{u}\times\mathbf{v} = \langle -2, 6, 8 \rangle,\quad \text{area} = 2 \sqrt{26} \)

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 to find the area of the parallelogram. The steps are logically sound and the final answer is correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the cross product and its magnitude to find the area of the parallelogram. The steps are logically sound and the final answer is correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to explicitly state the cross product vector \mathbf{u} \times \mathbf{v} = \langle -2, 6, 8 \rangle, which is the primary part of the question. It jumps directly to the magnitude calculation without presenting the vector result.
gpt-oss:20b: pass 2026-09-27

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 2026-09-27 with SymPy 1.14.0.