Cross product
Problem 9.147 · medium
Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle 4, 1, 0 \rangle,\ \mathbf{v} = \langle -2, 1, 4 \rangle \), and the area of the parallelogram they span.
- Expand the determinant with rows i j k, u, v.
- \[ \left[\begin{matrix}4\\-16\\6\end{matrix}\right] \]The components.
- \[ 2 \sqrt{77} \]Area = |u × v|.
Answer \( \mathbf{u}\times\mathbf{v} = \langle 4, -16, 6 \rangle,\quad \text{area} = 2 \sqrt{77} \)
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: fail (error) — The solution never actually computes the cross product; it merely repeats the same column vector. The determinant expansion step is not carried out, so the claimed result <4,-16,6> is unsupported.qwen3.6:27b-mlx: pass — The solution correctly computes the cross product and its magnitude. The sentences accurately describe the method used.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly computes the cross product and its magnitude. The sentences accurately describe the method used.gpt-oss:20b: fail (error) 2026-10-03 — The solution never actually computes the cross product; it merely repeats the same column vector. The determinant expansion step is not carried out, so the claimed result <4,-16,6> is unsupported.qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly computes the cross product and its magnitude to find the area of the parallelogram. 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 <4, -16, 6>, but the magnitude is sqrt(4^2 + (-16)^2 + 6^2) = sqrt(16 + 256 + 36) = sqrt(308) = 2*sqrt(77). Wait, let me re-calculate. 4i - 16j + 6k. Magnitude squared is 16 + 256 + 36 = 308. 308 = 4 * 77. So 2*sqrt(77) is correct. Why did I think it was wrong? Let me re-read the solution. The solution says the components are <4, -16, 6>. Let's check the cross product manually. u = <4, 1, 0>, v = <-2, 1, 4>. i(1*4 - 0*1) - j(4*4 - 0*(-2)) + k(4*1 - 1*(-2)) = i(4) - j(16) + k(4+2) = <4, -16, 6>. The magnitude is sqrt(16 + 256 + 36) = sqrt(308) = 2*sqrt(77). The solution is actually correct. Why would I flag it? Ah, looking at the provided solution text: 'Matrix([...]) ... -- The components.' It lists the vector twice? No, it just shows the vector. Is there a subtle error? The problem asks for u x v AND the area. The solution provides the vector components and the area. The area calculation is correct. The vector calculation is correct. Is there a style issue? The sentence 'Expand the determinant...' is a bit vague but acceptable. Let me look closer. Is the vector <4, -16, 6> correct? Yes. Is the area 2*sqrt(77) correct? Yes. So why 'fail'? Maybe I'm missing something. Let's re-read the prompt. 'Find u x v ... and the area'. The solution provides the vector and the area. It seems correct. Wait, looking at the JSON output format, I need to be adversarial. Is there a sign error? u x v = <4, -16, 6>. v x u = <-4, 16, -6>. The problem asks for u x v. The solution gives <4, -16, 6>. This is correct. The area is the magnitude. This is correct. Perhaps the 'error' is that the solution doesn't explicitly state the final vector answer in a standard form, just shows a matrix? Or maybe the 'unchecked' equation line is confusing? No, the instructions say equations are checked by CAS. The sentences are what I check. Sentence 1: 'Expand the determinant with rows i j k, u, v.' This is a valid method. Sentence 2 is an equation. Sentence 3 is an equation. There are no other sentences. The solution seems sound. Why would it fail? Let me double check the arithmetic one more time. u = <4, 1, 0>, v = <-2, 1, 4>. i component: 1*4 - 0*1 = 4. j component: -(4*4 - 0*(-2)) = -16. k component: 4*1 - 1*(-2) = 6. Vector: <4, -16, 6>. Magnitude: sqrt(16 + 256 + 36) = sqrt(308) = sqrt(4*77) = 2*sqrt(77). Everything is correct. I will mark it as pass.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.