Cross product
Problem 9.28 · medium
Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle 4, -2, 4 \rangle,\ \mathbf{v} = \langle -4, 4, 2 \rangle \), and the area of the parallelogram they span.
- Expand the determinant with rows i j k, u, v.
- \[ \left[\begin{matrix}-20\\-24\\8\end{matrix}\right] \]The components.
- \[ 4 \sqrt{65} \]Area = |u × v|.
Answer \( \mathbf{u}\times\mathbf{v} = \langle -20, -24, 8 \rangle,\quad \text{area} = 4 \sqrt{65} \)
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 solution fails to explicitly state the cross product vector \mathbf{u} \times \mathbf{v} = \langle -20, -24, 8 \rangle, which is the primary answer requested. It jumps directly to the magnitude calculation without presenting the vector result.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to explicitly state the cross product vector \mathbf{u} \times \mathbf{v} = \langle -20, -24, 8 \rangle, which is the primary answer requested. It jumps directly to the magnitude calculation without presenting the vector result.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the cross product and its magnitude. The steps are logically sound and the final answer matches the stated result.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly computes the cross product and its magnitude to find the area. 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.