Cross product
Problem 9.148 · easy
Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle 4, 0, -3 \rangle,\ \mathbf{v} = \langle -1, -4, -1 \rangle \), and the area of the parallelogram they span.
- Expand the determinant with rows i j k, u, v.Reviewed
- \[ \left[\begin{matrix}-12\\7\\-16\end{matrix}\right] \]The components.✓ Proved
- \[ \sqrt{449} \]Area = |u × v|.✓ Proved
Answer \( \mathbf{u}\times\mathbf{v} = \langle -12, 7, -16 \rangle,\quad \text{area} = \sqrt{449} \)
Lines: 2 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly applies the determinant formula for the cross product and the magnitude formula for the area. The steps are logically sound and the final answer is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the determinant formula for the cross product and the magnitude formula for the area. The steps are logically sound and the final answer is correct.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-28 — The cross product calculation is incorrect. The correct cross product is < -12, 7, -15 >, not < -12, 7, -16 >. The z-component should be (4)(-4) - (0)(-1) = -16? Wait, u_x*v_y - u_y*v_x = 4*(-4) - 0*(-1) = -16. Let me recheck. u = <4, 0, -3>, v = <-1, -4, -1>. i(0*-1 - (-3)*-4) - j(4*-1 - (-3)*-1) + k(4*-4 - 0*-1). i(0 - 12) - j(-4 - 3) + k(-16 - 0) = -12i + 7j - 16k. The vector is indeed <-12, 7, -16>. The magnitude squared is 144 + 49 + 256 = 449. The solution is actually correct. Why did I think it was wrong? Let me re-read the prompt. 'Expand the determinant...'. The result shown is correct. The area is sqrt(449). The solution is sound.gpt-oss:20b: pass 2026-09-28
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.