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Vectors in three dimensions

Problem 9.444 · medium

Find the vector \( \displaystyle \mathbf u \) with \( \displaystyle \|\mathbf u\| = 10 \) pointing in the same direction as \( \displaystyle \mathbf v = \left\langle 0, 0, -5 \right\rangle \).

\[ 5 \]
‖v‖.✓ Proved
\[ \left[\begin{matrix}0\\0\\-10\end{matrix}\right] \]
10·v/‖v‖.✓ Proved

Answer \( \left\langle 0, 0, -10 \right\rangle \)

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 computes the magnitude of v and scales it to the desired length, resulting in the correct vector.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly computes the magnitude of v and scales it to the desired length, resulting in the correct vector.
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution lacks any explanatory sentences describing the method (scaling the vector v by the ratio of the desired magnitude to its current magnitude). Without these sentences, the steps are just uncontextualized calculations, failing to model the problem-solving process.
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.