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Vectors in three dimensions

Problem 9.441 · medium

Find the vector \( \displaystyle \mathbf u \) with \( \displaystyle \|\mathbf u\| = 9 \) pointing in the same direction as \( \displaystyle \mathbf v = \left\langle 2, 1, 2 \right\rangle \).

\[ 3 \]
‖v‖.✓ Proved
\[ \left[\begin{matrix}6\\3\\6\end{matrix}\right] \]
9·v/‖v‖.✓ Proved

Answer \( \left\langle 6, 3, 6 \right\rangle \)

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 3 and scales v by 9/3 = 3 to obtain the correct vector <6, 3, 6>. The algebraic steps are verified.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly calculates the magnitude of v as 3 and scales v by 9/3 = 3 to obtain the correct vector <6, 3, 6>. The algebraic steps are verified.
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution incorrectly calculates the magnitude of v as 3 instead of 3, but then uses the correct magnitude in the final calculation without showing the work. Specifically, line 1 states '3 = 3 -- ||v||', which is numerically correct (sqrt(2^2+1^2+2^2) = sqrt(9) = 3), but the label is confusingly formatted. However, the main issue is that the solution provides no explanatory sentences, just two unchecked equations. The prompt requires reviewing sentences. Since there are no sentences to review for logical flow or setup, but the equations are marked 'unchecked' and the first one is trivial identity '3=3' labeled as norm, it's a bit ambiguous. Let's look closer. The problem asks to find u. The solution shows the norm calculation and the final answer. It misses the step of normalizing v. But more importantly, the prompt says 'Each line... is either an EQUATION... or a SENTENCE'. The provided solution has NO sentences. It just has two equations. The instructions say 'The sentences are what you are here for.' If there are no sentences, can I pass it? Usually, a solution with no text is poor style or incomplete. But let's look at the 'unchecked' tag. The user says 'a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. Here they are marked 'unchecked'. This implies I should check them? Or that they haven't been checked? The prompt says 'do not re-check the algebra of a line marked proved'. These are marked 'unchecked'. This is a contradiction in the prompt's meta-instructions vs the input. However, looking at the content: Line 1 says `3 = 3 -- ||v||`. This is a true statement. Line 2 says `Matrix(...) = Matrix(...) -- 9*v/||v||`. This is also a true statement. The answer is correct. Is it misleading? It skips the explanation of *why* we multiply by 9/||v||. A student might not learn the method. But is it an error? No. Is it misleading? Maybe. Is it style? Yes. But wait, the prompt says 'Find what is wrong'. If I have to find something wrong, the lack of explanation is the issue. However, often these 'adversarial' reviews look for mathematical errors. There are no mathematical errors in the equations provided. The norm is 3. The vector is <6,3,6>. The math is correct. The 'error' might be that the solution is just a list of equations without context. Let's look at the severity options. 'style' is for 'correct, but a sentence could be clearer'. Here there are no sentences. This is a 'style' issue because it's not a proper solution. Or is it 'misleading' because it doesn't teach the method? I will go with 'style' as the math is correct but the presentation is non-existent.
gpt-oss:20b: pass 2026-10-09

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 2026-10-09 with SymPy 1.14.0.