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Vectors in three dimensions

Problem 9.361 · medium

Find the vector \( \displaystyle \mathbf u \) with \( \displaystyle \|\mathbf u\| = 6 \) pointing in the same direction as \( \displaystyle \mathbf v = \left\langle 2, 1, 2 \right\rangle \).

\[ 3 \]
‖v‖.✓ Proved
\[ \left[\begin{matrix}4\\2\\4\end{matrix}\right] \]
6·v/‖v‖.✓ Proved

Answer \( \left\langle 4, 2, 4 \right\rangle \)

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 norm of v as 3 and scales v by 6/3 = 2 to obtain the correct vector <4, 2, 4>. The algebraic steps are verified.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly calculates the norm of v as 3 and scales v by 6/3 = 2 to obtain the correct vector <4, 2, 4>. The algebraic steps are verified.
qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly calculates the magnitude of v as 3 instead of 3. The vector v = <2, 1, 2> has magnitude sqrt(2^2 + 1^2 + 2^2) = sqrt(9) = 3
gpt-oss:20b: pass 2026-10-07

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 2026-10-07 with SymPy 1.14.0.