Cross product
Problem 9.21 · easy
Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle 4, -1, -2 \rangle,\ \mathbf{v} = \langle -1, 2, -1 \rangle \), and the area of the parallelogram they span.
- Expand the determinant with rows i j k, u, v.
- \[ \left[\begin{matrix}5\\6\\7\end{matrix}\right] \]The components.
- \[ \sqrt{110} \]Area = |u × v|.
Answer \( \mathbf{u}\times\mathbf{v} = \langle 5, 6, 7 \rangle,\quad \text{area} = \sqrt{110} \)
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 computed cross product components <5, 6, 7> are incorrect; the correct cross product is <-3, 6, 7>. Consequently, the stated area sqrt(110) is wrong (it should be sqrt(94)).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The computed cross product components <5, 6, 7> are incorrect; the correct cross product is <-3, 6, 7>. Consequently, the stated area sqrt(110) is wrong (it should be sqrt(94)).gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-27 — The computed cross product components <5, 6, 7> are incorrect; the correct cross product is <-3, 6, 7>. Consequently, the calculated area sqrt(110) is wrong; the correct area is sqrt(94).gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer returned a non-objectgpt-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.