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.Reviewed
- \[ \sqrt{449} \]Area = |u × v|.Reviewed
Answer \( \mathbf{u}\times\mathbf{v} = \langle -12, 7, -16 \rangle,\quad \text{area} = \sqrt{449} \)
Lines: 3 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. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 3 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the method for computing the cross product and the area of the parallelogram. The algebraic results are verified as correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies the method for computing the cross product and the area of the parallelogram. The algebraic results are verified as correct.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-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 2026-10-13 with SymPy 1.14.0.