Cross product
Problem 9.105 · easy
Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle -3, 0, 1 \rangle,\ \mathbf{v} = \langle -1, 0, -3 \rangle \), and the area of the parallelogram they span.
- Expand the determinant with rows i j k, u, v.
- \[ \left[\begin{matrix}0\\-10\\0\end{matrix}\right] \]The components.✓ Proved
- \[ 10 \]Area = |u × v|.✓ Proved
Answer \( \mathbf{u}\times\mathbf{v} = \langle 0, -10, 0 \rangle,\quad \text{area} = 10 \)
Lines: 2 proved, 1 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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | 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: fail (error) — The solution repeats the cross‑product vector twice and then lists the area as "10, 10" instead of a single value 10. The correct cross product is (0, −10, 0) and the area of the parallelogram is |(0, −10, 0)| = 10.qwen3.6:27b-mlx: fail (error) — The solution claims the cross product components are <0, -10, 0>, but the correct calculation for u x v yields <0, -10, 0> is incorrect; the y-component is (-3)(-3) - (1)(-1) = 10, not -10. Wait, let me re-calculate. u = <-3, 0, 1>, v = <-1, 0, -3>. i(0*-3 - 1*0) - j(-3*-3 - 1*-1) + k(-3*0 - 0*-1) = i(0) - j(9 + 1) + k(0) = <0, -10, 0>. The cross product is indeed <0, -10, 0>. The magnitude is 10. The solution is actually correct. Why did I think it was wrong? Let me re-read the prompt. 'Find u x v ... and the area'. The stated answer is 10. The solution calculates the cross product vector and then the area. The equation line says 'Matrix([...]) ... -- The components'. It shows the vector <0, -10, 0>. Then it calculates area 10. This is correct. Is there a subtle error? The prompt asks for 'u x v' AND 'the area'. The stated answer is just '10'. Usually, if a problem asks for two things, the answer should reflect both, or the 'stated answer' refers to the final numerical value requested if it's a single-box input. However, the solution text itself is mathematically sound. Let's look closer at the 'equation' line. It shows two identical matrices. This is likely a formatting artifact of the CAS output showing the result twice or comparing it. It doesn't constitute a mathematical error in the derivation. The sentence 'Expand the determinant...' is a valid method. The calculation is correct. The area is correct. I will mark it as pass.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution claims the cross product components are <0, -10, 0>, but the correct calculation for u x v yields <0, -10, 0> is incorrect; the y-component is (-3)(-3) - (1)(-1) = 10, not -10. Wait, let me re-calculate. u = <-3, 0, 1>, v = <-1, 0, -3>. i(0*-3 - 1*0) - j(-3*-3 - 1*-1) + k(-3*0 - 0*-1) = i(0) - j(9 + 1) + k(0) = <0, -10, 0>. The cross product is indeed <0, -10, 0>. The magnitude is 10. The solution is actually correct. Why did I think it was wrong? Let me re-read the prompt. 'Find u x v ... and the area'. The stated answer is 10. The solution calculates the cross product vector and then the area. The equation line says 'Matrix([...]) ... -- The components'. It shows the vector <0, -10, 0>. Then it calculates area 10. This is correct. Is there a subtle error? The prompt asks for 'u x v' AND 'the area'. The stated answer is just '10'. Usually, if a problem asks for two things, the answer should reflect both, or the 'stated answer' refers to the final numerical value requested if it's a single-box input. However, the solution text itself is mathematically sound. Let's look closer at the 'equation' line. It shows two identical matrices. This is likely a formatting artifact of the CAS output showing the result twice or comparing it. It doesn't constitute a mathematical error in the derivation. The sentence 'Expand the determinant...' is a valid method. The calculation is correct. The area is correct. I will mark it as pass.gpt-oss:20b: fail (error) 2026-09-26 — The solution repeats the cross‑product vector twice and then lists the area as "10, 10" instead of a single value 10. The correct cross product is (0, −10, 0) and the area of the parallelogram is |(0, −10, 0)| = 10.qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer returned a non-objectgpt-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.