Home›Calculus 3›Cross product›Problem 9.195

Cross product

Problem 9.195 · medium

Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle 4, -1, 1 \rangle,\ \mathbf{v} = \langle 0, -1, 1 \rangle \), and the area of the parallelogram they span.

Expand the determinant with rows i j k, u, v.
Reviewed
\[ \left[\begin{matrix}0\\-4\\-4\end{matrix}\right] \]
The components.✓ Proved
\[ 4 \sqrt{2} \]
Area = |u × v|.✓ Proved

Answer \( \mathbf{u}\times\mathbf{v} = \langle 0, -4, -4 \rangle,\quad \text{area} = 4 \sqrt{2} \)

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: pass
qwen3.6:27b-mlx: pass — The solution correctly computes the cross product and the area of the parallelogram. The steps are logically sound and the final answer matches the stated answer.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly computes the cross product and the area of the parallelogram. The steps are logically sound and the final answer matches the stated answer.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The cross product calculation is incorrect. The correct cross product is <0, -4, -4> is wrong; it should be <0, -4, -4>? Let's recheck: u=<4,-1,1>, v=<0,-1,1>. i(-1*1 - 1*-1) - j(4*1 - 1*0) + k(4*-1 - -1*0) = i(-1+1) - j(4) + k(-4) = <0, -4, -4>. Wait, the stated answer says <0, -4, -4>. Let me re-read the prompt's stated answer. Stated answer: <0, -4, -4>. My calculation: <0, -4, -4>. They match. Why did I think it was wrong? Let me re-calculate carefully. u x v = det|i j k; 4 -1 1; 0 -1 1| = i((-1)(1) - (1)(-1)) - j((4)(1) - (1)(0)) + k((4)(-1) - (-1)(0)) = i(-1 + 1) - j(4) + k(-4) = 0i - 4j - 4k = <0, -4, -4>. The stated answer is correct. The solution steps are just placeholders. Is there an error in the solution text? Step 1 says 'Expand the determinant...'. Step 2 shows the result. Step 3 shows the area. The area is sqrt(0^2 + (-4)^2 + (-4)^2) = sqrt(32) = 4*sqrt(2). The solution is mathematically correct. However, the prompt asks me to review 'worked calculus solutions adversarially'. The solution provided is extremely sparse. It doesn't show the work. But is it 'wrong'? No. Is it 'misleading'? Maybe, because it doesn't show the steps. But the instructions say 'Each line... is either an EQUATION... or a SENTENCE'. The sentences are 'Expand the determinant...'. This is a valid instruction. The equations are checked. Wait, look at the equation lines. They are marked 'unchecked' in the prompt description but the instructions say 'a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. The prompt says 'unchecked' in the text provided by the user, but the system prompt says 'do not re-check the algebra of a line marked proved'. The user's input says '[equation, unchecked]'. This might be a trick. If the equation is unchecked, I should check it? No, the system prompt says 'a computer algebra system has already checked... do not re-check...'. The 'unchecked' tag in the user input might be metadata I should ignore or it might mean the CAS didn't check it? The system prompt says 'Each line... is either an EQUATION, which a computer algebra system has already checked'. This implies I should trust the equations. The sentences are what I check. Sentence 1: 'Expand the determinant with rows i j k, u, v.' This is a correct method. Sentence 2 is an equation. Sentence 3 is an equation. There are no other sentences. The solution is technically sound but very brief. Is there a subtle error? Let's look at the cross product again. u=<4, -1, 1>, v=<0, -1, 1>. i component: (-1)(1) - (1)(-1) = -1 + 1 = 0. j component: -(4(1) - 1(0)) = -4. k component: 4(-1) - (-1)(0) = -4. Result <0, -4, -4>. Area sqrt(16+16) = sqrt(32) = 4sqrt(2). Everything is correct. Why would it fail? Maybe the 'unchecked' tag implies I should check it? If I check it, it's correct. Maybe the issue is that the solution doesn't explicitly state the formula for area? Step 3 says 'Area = |u x v|' in the comment. That's fine. I will pass it.
gpt-oss:20b: pass 2026-10-03

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 2026-10-03 with SymPy 1.14.0.