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Vectors in three dimensions

Problem 9.604 · medium

Find the vector \( \displaystyle \mathbf u \) with \( \displaystyle \|\mathbf u\| = 3 \) pointing in the same direction as \( \displaystyle \mathbf v = \left\langle 2, 6, -9 \right\rangle \).

\[ 11 \]
‖v‖.✓ Proved
\[ \left[\begin{matrix}\frac{6}{11}\\\frac{18}{11}\\- \frac{27}{11}\end{matrix}\right] \]
3·v/‖v‖.✓ Proved

Answer \( \left\langle \frac{6}{11}, \frac{18}{11}, - \frac{27}{11} \right\rangle \)

Every line of this solution was proved by the computer algebra system SymPy. 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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	✓ 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	length checked, and parallel by the equality case of Cauchy–Schwarz

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly calculates the magnitude of v as 11 and scales the vector by 3/11 to achieve the desired magnitude of 3 while maintaining direction.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-13
qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly calculates the magnitude of v as 11 and scales the vector by 3/11 to achieve the desired magnitude of 3 while maintaining direction.
qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution claims the magnitude of v is 11, but the actual magnitude is sqrt(4 + 36 + 81) = sqrt(121) = 11. Wait, 2^2 + 6^2 + (-9)^2 = 4 + 36 + 81 = 121. sqrt(121) is indeed 11. The calculation is correct. Let me re-read the prompt constraints. 'Each line of the solution is either an EQUATION... or a SENTENCE'. The provided solution only contains equations marked as 'unchecked'. The prompt asks me to review 'worked calculus solutions'. Usually, there are sentences explaining the steps. Here, there are no sentences to review for logical flow, only equations. However, the prompt says 'The sentences are what you are here for.' If there are no sentences, do I pass it? Or is the lack of explanation a style issue? Or is the fact that the equations are 'unchecked' a problem? 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)'. These are marked 'unchecked'. This implies I should check them? No, it says 'do not re-check the algebra of a line marked proved'. These are NOT marked proved. They are marked 'unchecked'. This is ambiguous. Let's look at the content. The math is correct. The vector u is 3 * v / ||v||. ||v|| = 11. u = 3/11 * <2, 6, -9> = <6/11, 18/11, -27/11>. The answer is correct. The equations are correct. There are no sentences to critique. Is the absence of sentences a failure? The prompt says 'Review it.' and 'The sentences are what you are here for.' If there are no sentences, perhaps it's a 'style' fail for lacking explanation, or a 'pass' because the math is right. But usually, these adversarial reviews look for logical errors in the text. With no text, there are no logical errors in the text. However, looking closely at the first equation: '11 = 11 -- ||v||'. This is a tautology, not a calculation. It doesn't show *why* ||v|| is 11. It just asserts it. A student might not know where 11 came from. This is 'misleading' or 'style'. But is it an 'error'? No. Is it 'misleading'? It hides the work. Let's look for a deeper error. Did I calculate ||v|| correctly? 2^2=4, 6^2=36, (-9)^2=81. 4+36=40. 40+81=121. sqrt(121)=11. Yes. The final vector is correct. The only issue is the lack of explanatory text or the tautological nature of the first equation. Given the strict instruction 'The sentences are what you are here for', and there are none, I will flag it as a style issue for lacking the necessary explanatory steps that constitute a 'solution' rather than just an answer key.
gpt-oss:20b: pass 2026-10-13

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/vectors_space, checked 2026-10-13 with SymPy 1.14.0.