Cross product
Problem 9.27 · medium
Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle -2, -2, 2 \rangle,\ \mathbf{v} = \langle 0, 1, 0 \rangle \), and the area of the parallelogram they span.
- Expand the determinant with rows i j k, u, v.
- \[ \left[\begin{matrix}-2\\0\\-2\end{matrix}\right] \]The components.
- \[ 2 \sqrt{2} \]Area = |u × v|.
Answer \( \mathbf{u}\times\mathbf{v} = \langle -2, 0, -2 \rangle,\quad \text{area} = 2 \sqrt{2} \)
Lines: 3 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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: fail (error) — The solution incorrectly lists the matrix for the cross product: it repeats vector u twice and omits v. The cross‑product computation itself is correct, but the presented equations do not reflect the proper setup.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the cross product vector, which is the first part of the problem. Furthermore, the intermediate equation line is malformed and does not clearly show the calculation of the cross product components, making the derivation of the area opaque.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to explicitly state the cross product vector, which is the first part of the problem. Furthermore, the intermediate equation line is malformed and does not clearly show the calculation of the cross product components, making the derivation of the area opaque.gpt-oss:20b: fail (error) 2026-09-28 — The solution incorrectly lists the matrix for the cross product: it repeats vector u twice and omits v. The cross‑product computation itself is correct, but the presented equations do not reflect the proper setup.qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to explicitly state the cross product vector \mathbf{u} \times \mathbf{v} = \langle -2, 0, -2 \rangle, which is the primary part of the question. It jumps directly to the magnitude calculation without presenting the vector result.gpt-oss:20b: fail (error) 2026-09-27 — The matrix in step 2 incorrectly lists the same vector twice and omits the correct components of υ and β. It does not represent the cross‑product determinant, so the subsequent area calculation is invalid.qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to explicitly state the cross product vector \mathbf{u} \times \mathbf{v} = \langle -2, 0, -2 \rangle, which is the primary part of the question. It jumps directly to the magnitude calculation without presenting the vector result.gpt-oss:20b: fail (error) 2026-09-26 — The matrix in step 2 is incorrect: it repeats the vector 'u' twice and omits 'v'. The cross‑product computation is not shown, yet the final magnitude is correct. The setup and conclusion are fine, but the intermediate statement is false.
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.