Cross product
Problem 9.102 · easy
Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle -2, 0, 1 \rangle,\ \mathbf{v} = \langle 0, -1, 0 \rangle \), and the area of the parallelogram they span.
- Expand the determinant with rows i j k, u, v.
- \[ \left[\begin{matrix}1\\0\\2\end{matrix}\right] \]The components.
- \[ \sqrt{5} \]Area = |u × v|.
Answer \( \mathbf{u}\times\mathbf{v} = \langle 1, 0, 2 \rangle,\quad \text{area} = \sqrt{5} \)
Lines: 3 not checked. 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 | Not checked | — | a sentence; read, not computed |
| 2 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 3 | Not checked | — | 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: passqwen3.6:27b-mlx: fail (error) — The solution fails to show the calculation of the cross product components, jumping directly to the final area. While the final area is correct, the intermediate step (the cross product vector) is missing, and the 'proved' equation lines are nonsensical placeholders rather than valid mathematical steps.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to show the calculation of the cross product components, jumping directly to the final area. While the final area is correct, the intermediate step (the cross product vector) is missing, and the 'proved' equation lines are nonsensical placeholders rather than valid mathematical steps.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to explicitly state the cross product vector, which is required by the problem statement ('Find u x v'). It jumps directly to the magnitude calculation without showing the intermediate vector result.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to actually compute the cross product, providing no intermediate steps or result for \mathbf{u} \times \mathbf{v}. Furthermore, the unchecked equations contain nonsensical data (identical matrices and repeated scalars) that do not correspond to the vectors given in the problem.gpt-oss:20b: inconclusive 2026-09-26 — reviewer returned a non-object
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.