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Vectors in three dimensions

Problem 9.523 · medium

Find the vector \( \displaystyle \mathbf u \) with \( \displaystyle \|\mathbf u\| = 6 \) pointing in the same direction as \( \displaystyle \mathbf v = \left\langle 0, 0, 5 \right\rangle \).

\[ 5 \]
‖v‖.✓ Proved
\[ \left[\begin{matrix}0\\0\\6\end{matrix}\right] \]
6·v/‖v‖.✓ Proved

Answer \( \left\langle 0, 0, 6 \right\rangle \)

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: fail (error) — The solution fails to explicitly state the formula for scaling a vector to a specific magnitude, making the transition from the problem statement to the final calculation opaque. It also presents the final answer as a matrix rather than a vector, which is a notational inconsistency.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to explicitly state the formula for scaling a vector to a specific magnitude, making the transition from the problem statement to the final calculation opaque. It also presents the final answer as a matrix rather than a vector, which is a notational inconsistency.
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution lacks any explanatory text or sentences to justify the steps. It presents raw equations without stating the method (normalizing the vector) or connecting the given vector v to the final answer, failing to model the problem-solving process.

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 2026-10-11 with SymPy 1.14.0.