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Cross product

Problem 9.175 · easy

Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle 4, -1, -4 \rangle,\ \mathbf{v} = \langle 4, 2, 3 \rangle \), and the area of the parallelogram they span.

Expand the determinant with rows i j k, u, v.
Reviewed
\[ \left[\begin{matrix}5\\-28\\12\end{matrix}\right] \]
The components.✓ Proved
\[ \sqrt{953} \]
Area = |u × v|.✓ Proved

Answer \( \mathbf{u}\times\mathbf{v} = \langle 5, -28, 12 \rangle,\quad \text{area} = \sqrt{953} \)

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 identifies the method for computing the cross product and the area of the parallelogram. The intermediate vector and final scalar result are correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the method for computing the cross product and the area of the parallelogram. The intermediate vector and final scalar result are correct.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The cross product calculation is incorrect. The correct cross product is <5, -28, 12>, but the solution claims the result is <5, -28, 12> while the algebra check shows the components are actually <5, -28, 12>... wait, let me re-calculate. u = <4, -1, -4>, v = <4, 2, 3>. i(-1*3 - (-4)*2) = i(-3 + 8) = 5i. j(4*3 - (-4)*4) = j(12 + 16) = 28j, but with minus sign from determinant expansion, it is -28j. k(4*2 - (-1)*4) = k(8 + 4) = 12k. So <5, -28, 12> is actually correct. Why did I think it was wrong? Let me re-read the prompt's provided solution lines. Line 2 says the components are <5, -28, 12>. Line 3 says area is sqrt(953). 5^2 + (-28)^2 + 12^2 = 25 + 784 + 144 = 953. The calculation is actually correct. However, the prompt says 'Each line... is either an EQUATION... or a SENTENCE'. Line 1 is a sentence. Line 2 is an equation (unchecked). Line 3 is an equation (unchecked). The review instructions say 'do not re-check the algebra of a line marked proved'. But these are marked 'unchecked'. Wait, the prompt says 'a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. The lines are marked 'unchecked'. This implies I SHOULD check them? Or does 'unchecked' mean the CAS didn't check it? The prompt says 'Each line... is either an EQUATION, which a computer algebra system has already checked...'. This implies all equations ARE checked. The label 'unchecked' in the text might be a distractor or a typo in the prompt format description vs the actual data. Let's look closer. '2. [equation, unchecked] ... -- The components.' If the CAS checked it, and it says 'unchecked', maybe it means the CAS output is shown but not verified by a human? No, the instruction says 'do not re-check the algebra of a line marked proved'. It doesn't say 'do not re-check unchecked lines'. It says 'Each line... is either an EQUATION, which a computer algebra system has already checked'. This is a definition of the line type. So ALL equations are checked by CAS. The tag 'unchecked' might just be metadata from the source. If the CAS checked it, and the result is <5, -28, 12>, then the algebra is correct. My manual calculation confirms <5, -28, 12>. So the math is correct. Is there a sentence error? 'Expand the determinant with rows i j k, u, v.' This is a valid method. 'Area = |u x v|'. This is a valid theorem. The conclusion follows. So why would it fail? Let me re-read carefully. u = <4, -1, -4>, v = <4, 2, 3>. i component: (-1)(3) - (-4)(2) = -3 + 8 = 5. j component: -( (4)(3) - (-4)(4) ) = -(12 + 16) = -28. k component: (4)(2) - (-1)(4) = 8 + 4 = 12. Result <5, -28, 12>. Magnitude sqrt(25 + 784 + 144) = sqrt(953). Everything is correct. Wait, is there a subtle issue? 'Expand the determinant with rows i j k, u, v.' Standard notation. Maybe the issue is that the solution doesn't explicitly state the cross product vector in the final answer, only the area? The problem asks for 'u x v ... AND the area'. The solution provides the components in line 2 and the area in line 3. It doesn't explicitly state 'u x v = <5, -28, 12>' as a final conclusion, but line 2 shows the components. Is that sufficient? The prompt asks to review the solution. If the solution is correct, I should pass. But I am an adversarial reviewer. Is there anything wrong? The problem asks for TWO things: the cross product and the area. The solution calculates both. Line 2 gives the vector. Line 3 gives the area. It seems complete. Let me check the severity definitions. 'error' if a step is wrong. 'misleading' if a student learns something false. 'style' if unclear. If it's correct, 'pass'. I will double check the arithmetic one more time. 5^2=25. 28^2=784. 12^2=144. 25+784=809. 809+144=953. Correct. The cross product components are correct. The method is correct. I see no error. I will pass.
gpt-oss:20b: pass 2026-09-29

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 2026-09-29 with SymPy 1.14.0.