Cross product
Problem 9.70 · easy
Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle -2, -3, 0 \rangle,\ \mathbf{v} = \langle 3, 1, -3 \rangle \), and the area of the parallelogram they span.
- Expand the determinant with rows i j k, u, v.Reviewed
- \[ \left[\begin{matrix}9\\-6\\7\end{matrix}\right] \]The components.Reviewed
- \[ \sqrt{166} \]Area = |u × v|.Reviewed
Answer \( \mathbf{u}\times\mathbf{v} = \langle 9, -6, 7 \rangle,\quad \text{area} = \sqrt{166} \)
✓ 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 applies the definition of the cross product via determinant expansion and the geometric interpretation of the cross product magnitude as the area of the parallelogram. The calculations are verified as correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the definition of the cross product via determinant expansion and the geometric interpretation of the cross product magnitude as the area of the parallelogram. The calculations are verified as correct.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the cross product and its magnitude. The steps are logically sound and the final answer matches the stated result.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the cross product components and the area formula. The steps are logically sound and the calculations are verified.gpt-oss:20b: pass 2026-09-26
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.