Cross product
Problem 9.150 · easy
Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle 1, -4, -1 \rangle,\ \mathbf{v} = \langle 0, -1, 0 \rangle \), and the area of the parallelogram they span.
- Expand the determinant with rows i j k, u, v.
- \[ \left[\begin{matrix}-1\\0\\-1\end{matrix}\right] \]The components.✓ Proved
- \[ \sqrt{2} \]Area = |u × v|.✓ Proved
Answer \( \mathbf{u}\times\mathbf{v} = \langle -1, 0, -1 \rangle,\quad \text{area} = \sqrt{2} \)
Lines: 2 proved, 1 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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | 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: fail (error) — The cross product calculation is incorrect; the determinant expansion yields < -1, 0, -1 >, but the solution presents the vector as < -1, 0, -1 > in a confusing matrix format and fails to explicitly state the vector result before calculating magnitude. More critically, the magnitude of < -1, 0, -1 > is sqrt(2), which is correct, but the intermediate step is presented as two identical matrices rather than a vector, making the logic opaque and potentially misleading about the structure of the cross product result.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The cross product calculation is incorrect; the determinant expansion yields < -1, 0, -1 >, but the solution presents the vector as < -1, 0, -1 > in a confusing matrix format and fails to explicitly state the vector result before calculating magnitude. More critically, the magnitude of < -1, 0, -1 > is sqrt(2), which is correct, but the intermediate step is presented as two identical matrices rather than a vector, making the logic opaque and potentially misleading about the structure of the cross product result.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to compute the cross product components correctly. The determinant expansion yields < -1, 0, -1 >, but the provided matrix data is ambiguous and does not clearly show the calculation steps. More importantly, the final area calculation is correct (sqrt(2)), but the intermediate step showing the cross product vector is presented as a matrix equality that is syntactically confusing and potentially misleading about the vector result. However, the primary issue is that the 'equation' lines are not standard mathematical statements but rather data dumps, making the 'sentence' claim of expanding the determinant unsupported by clear algebraic work in the provided lines. Actually, looking closer, the cross product is < -1, 0, -1 >. The magnitude is sqrt(1+0+1) = sqrt(2). The answer is correct. But the solution text is garbage. Line 2 shows two identical matrices? That's wrong. It should show the vector. Line 2 is nonsensical. Therefore, the solution is flawed.gpt-oss:20b: inconclusive 2026-09-28 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution incorrectly lists the components of \u and \v as \langle-1,0,-1\rangle, which does not match the given vectors. The cross product and area ar
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 2026-09-28 with SymPy 1.14.0.