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.
- \[ 10 \]Area = |u × v|.
Answer \( \mathbf{u}\times\mathbf{v} = \langle 0, -10, 0 \rangle,\quad \text{area} = 10 \)
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: passqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the resulting vector \mathbf{u} \times \mathbf{v} = \langle 0, -10, 0 \rangle, which is the primary part of the question. While the magnitude calculation is correct, omitting the vector result makes the solution incomplete and potentially misleading regarding the cross product's direction.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to explicitly state the resulting vector \mathbf{u} \times \mathbf{v} = \langle 0, -10, 0 \rangle, which is the primary part of the question. While the magnitude calculation is correct, omitting the vector result makes the solution incomplete and potentially misleading regarding the cross product's direction.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-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 2026-10-13 with SymPy 1.14.0.