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.✓ Proved
- \[ 6 \sqrt{5} \]Area = |u × v|.✓ Proved
Answer \( \mathbf{u}\times\mathbf{v} = \langle 12, -6, 0 \rangle,\quad \text{area} = 6 \sqrt{5} \)
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: passqwen3.6:27b-mlx: pass — The solution correctly computes the cross product and its magnitude. The steps are logically sound and the final answer is correct.
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 is correct.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-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 2026-09-28 with SymPy 1.14.0.