Cross product
Problem 9.198 · medium
Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle 3, 2, 2 \rangle,\ \mathbf{v} = \langle 3, 4, 0 \rangle \), and the area of the parallelogram they span.
- Expand the determinant with rows i j k, u, v.Reviewed
- \[ \left[\begin{matrix}-8\\6\\6\end{matrix}\right] \]The components.✓ Proved
- \[ 2 \sqrt{34} \]Area = |u × v|.✓ Proved
Answer \( \mathbf{u}\times\mathbf{v} = \langle -8, 6, 6 \rangle,\quad \text{area} = 2 \sqrt{34} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 sets up the cross product via the determinant method and correctly identifies the area as the magnitude of the resulting vector. The algebraic results are verified.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly sets up the cross product via the determinant method and correctly identifies the area as the magnitude of the resulting vector. The algebraic results are verified.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies the method for computing the cross product and the area of the parallelogram. The algebraic results are correct, and the logical steps are sound.gpt-oss:20b: pass 2026-10-03
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 2026-10-03 with SymPy 1.14.0.