Cross product
Problem 9.106 · medium
Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle -2, -3, -4 \rangle,\ \mathbf{v} = \langle 4, 2, 0 \rangle \), and the area of the parallelogram they span.
- Expand the determinant with rows i j k, u, v.Reviewed
- \[ \left[\begin{matrix}8\\-16\\8\end{matrix}\right] \]The components.✓ Proved
- \[ 8 \sqrt{6} \]Area = |u × v|.✓ Proved
Answer \( \mathbf{u}\times\mathbf{v} = \langle 8, -16, 8 \rangle,\quad \text{area} = 8 \sqrt{6} \)
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 identifies the method for finding the cross product and the area of the parallelogram. The calculations are verified as correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the method for finding the cross product and the area of the parallelogram. The calculations are verified as correct.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (error) 2026-09-26 — The cross product calculation is incorrect; the correct result is <8, -16, 8>, but the magnitude is sqrt(8^2 + (-16)^2 + 8^2) = sqrt(384) = 8*sqrt(6). Wait, let me re-calculate. u = <-2, -3, -4>, v = <4, 2, 0>. i(-3*0 - (-4)*2) - j(-2*0 - (-4)*4) + k(-2*2 - (-3)*4) = i(8) - j(16) + k(-4+12) = <8, -16, 8>. The vector in line 2 is correct. The magnitude is sqrt(64 + 256 + 64) = sqrt(384) = sqrt(64*6) = 8*sqrt(6). The solution is actually correct. Why did I think it was wrong? Let me check the prompt again. 'Find u x v ... and the area'. The solution provides the vector components in line 2 and the area in line 3. The vector is <8, -16, 8>. The area is 8*sqrt(6). The stated answer is 8*sqrt(6). The solution seems correct. Is there a subtle error? Line 1 says 'Expand the determinant...'. Line 2 shows the result. Line 3 shows the area. It seems fine. Let me look closer at the vector. i component: (-3)(0) - (-4)(2) = 8. j component: -((-2)(0) - (-4)(4)) = -(16) = -16. k component: (-2)(2) - (-3)(4) = -4 + 12 = 8. Vector is <8, -16, 8>. Magnitude sqrt(64+256+64) = sqrt(384) = 8sqrt(6). Everything is correct. I will mark it as pass.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 2026-09-26 with SymPy 1.14.0.