Cross product
Problem 9.171 · easy
Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle 1, -2, 0 \rangle,\ \mathbf{v} = \langle -1, 1, 0 \rangle \), and the area of the parallelogram they span.
- Expand the determinant with rows i j k, u, v.Reviewed
- \[ \left[\begin{matrix}0\\0\\-1\end{matrix}\right] \]The components.Reviewed
- \[ 1 \]Area = |u × v|.Reviewed
Answer \( \mathbf{u}\times\mathbf{v} = \langle 0, 0, -1 \rangle,\quad \text{area} = 1 \)
✓ Nihil obstat Lines: 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the method for computing the cross product and the area. The algebraic results are verified as correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the method for computing the cross product and the area. The algebraic results are verified as correct.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution fails to state the final vector result for u x v, which is a required part of the problem. Additionally, the area calculation is incorrect: |<0, 0, -1>| = 1, but the problem asks for the area of the parallelogram, which is indeed 1, so the area part is correct, but the missing vector output makes the solution incomplete and thus fails to answer the full prompt.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: fail (error) 2026-09-29 — The cross product calculation is incorrect; the result should be <0, 0, -1>, not <0, 0, 1> (or whatever was implied by the matrix output which seems malformed). Consequently, the area calculation is based on a wrong vector, although the magnitude happens to be 1 in this specific case, the intermediate vector is wrong. More importantly, the solution format is garbled and does not clearly show the correct cross product vector <0, 0, -1>.gpt-oss:20b: pass 2026-09-29
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.