Cross product
Problem 9.16 · easy
Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle -1, 0, 1 \rangle,\ \mathbf{v} = \langle -2, 4, 1 \rangle \), and the area of the parallelogram they span.
- Expand the determinant with rows i j k, u, v.
- \[ \left[\begin{matrix}-4\\-1\\-4\end{matrix}\right] \]The components.
- \[ \sqrt{33} \]Area = |u × v|.
Answer \( \mathbf{u}\times\mathbf{v} = \langle -4, -1, -4 \rangle,\quad \text{area} = \sqrt{33} \)
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 cross product calculation is incorrect. The correct cross product is <4, 3, -4>, not <-4, -1, -4>. Consequently, the area is sqrt(41), not sqrt(33).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The cross product calculation is incorrect. The correct cross product is <4, 3, -4>, not <-4, -1, -4>. Consequently, the area is sqrt(41), not sqrt(33).gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the cross product components and the magnitude. The steps are logically sound and the final answer is correct.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — 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 2026-09-26
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.