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Cross product

Problem 9.62 · easy

Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle -4, 3, 0 \rangle,\ \mathbf{v} = \langle 4, -1, 3 \rangle \), and the area of the parallelogram they span.

Expand the determinant with rows i j k, u, v.
Reviewed
\[ \left[\begin{matrix}9\\12\\-8\end{matrix}\right] \]
The components.Reviewed
\[ 17 \]
Area = |u × v|.Reviewed

Answer \( \mathbf{u}\times\mathbf{v} = \langle 9, 12, -8 \rangle,\quad \text{area} = 17 \)

Lines: 3 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. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass
qwen3.6:27b-mlx: pass — The solution correctly computes the cross product and its magnitude. The final answer matches the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly computes the cross product and its magnitude. The final answer matches the stated answer.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the cross product and its magnitude. The stated answer matches the calculated area.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution claims the area is 17, but the magnitude of the cross product vector <9, 12, -8> is sqrt(81 + 144 + 64) = sqrt(289) = 17. While the final number is correct, the solution fails to show the calculation of the magnitude, jumping from the vector components directly to the scalar 17 without stating the formula |u x v| = sqrt(x^2 + y^2 + z^2). This is a significant omission in a calculus solution where showing work is expected. However, looking closer at the prompt's constraints: 'Each line... is either an EQUATION... or a SENTENCE'. Line 3 is marked [equation, proved] with result '17, 17'. The text says 'Area = |u x v|'. If the CAS checked that |<9,12,-8>| = 17, then the equation is correct. But is the *sentence* or the *logic* sound? The prompt asks to review sentences. Line 1 is a sentence. Line 3 is an equation. Wait, the prompt says 'The sentences are what you are here for.' but also 'Check that... each conclusion follows from the lines before it'. Line 3 is an equation. If it is marked proved, I should not re-check the algebra. But does the solution *state* the method? Line 1 says 'Expand the determinant...'. It does not explicitly state 'Calculate the magnitude of the resulting vector'. However, Line 3 is an equation. If the equation is proved, it is correct. Is there a sentence error? Line 1 is just 'Expand the determinant...'. It doesn't mention the area part. The problem asks for two things: u x v and the area. Line 2 gives u x v. Line 3 gives the area. The transition from 2 to 3 is implicit. Is there a sentence that is wrong? No explicit sentence claims the area formula. But usually, 'Expand the determinant' refers to the cross product. The area calculation is a separate step. If the solution is just lines 1-3, and 3 is an equation, then there are no sentences describing the area calculation. Is this 'misleading' or 'style'? Or is it an 'error' because the step is missing? The prompt says 'Check that... each conclusion follows from the lines before it'. If line 3 is an equation, its truth is guaranteed by 'proved'. But does it *follow* from line 2? In a rigorous sense, yes, if we assume standard definitions. But usually, we need to state 'The area is the magnitude of the cross product'. Without that sentence, a student might not know *why* we are taking the magnitude. However, the prompt says 'The sentences are what you are here for.' If there are no sentences about the area, maybe I should check if the existing sentences are correct. Line 1 is correct. Line 3 is an equation. If the verdict is 'pass', notes are empty. If 'fail', I must find a sentence error. There are no sentence errors in the text provided, only a missing explanation. But wait, look at Line 3: '17, 17 -- Area = |u x v|'. This is marked as an equation. The comment explains it. Is the comment a sentence? No, it's a comment on the equation. The prompt says 'Each line... is either an EQUATION... or a SENTENCE'. Line 3 is an equation. So I don't review the comment as a sentence? Or is the comment part of the 'sentence' category? Usually, comments are ignored or treated as context. If I treat the comment as a sentence, it is correct. If I treat it as a comment, there is no sentence for the area step. This is a 'style' issue or 'misleading' if the student doesn't see the link. But 'error' is for wrong steps. The step is correct. Let's look for a subtle error. u = <-4, 3, 0>, v = <4, -1, 3>. i(9-0) - j(-12-0) + k(4-12) = 9i + 12j - 8k. Correct. Magnitude sqrt(81+144+64) = sqrt(289) = 17. Correct. The solution is mathematically sound. The only issue is brevity. Is brevity a 'style' fail? 'style (correct, but a sentence could be clearer)'. Yes. The connection between the cross product and the area is not explicitly stated in a sentence, which is poor pedagogical style.
gpt-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.