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Cross product

Problem 9.153 · medium

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

Expand the determinant with rows i j k, u, v.
Reviewed
\[ \left[\begin{matrix}12\\-6\\0\end{matrix}\right] \]
The components.Reviewed
\[ 6 \sqrt{5} \]
Area = |u × v|.Reviewed

Answer \( \mathbf{u}\times\mathbf{v} = \langle 12, -6, 0 \rangle,\quad \text{area} = 6 \sqrt{5} \)

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 its magnitude. The steps are logically sound and the results match the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly computes the cross product and its magnitude. The steps are logically sound and the results match 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 is correct.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The cross product calculation is incorrect. The z-component should be (-1)(-4) - (-2)(-2) = 4 - 4 = 0, but the x and y components are also wrong: i(-4*2 - 4*(-4)) = i(-8+16) = 8i, and j(-((-1)*2 - 4*(-2))) = j(-(-2+8)) = -6j. The correct cross product is <8, -6, 0>, not <12, -6, 0>. Consequently, the magnitude is sqrt(64+36) = 10, not 6*sqrt(5).
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.