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Vectors in three dimensions

Problem 9.281 · medium

Find the vector \( \displaystyle \mathbf u \) with \( \displaystyle \|\mathbf u\| = 10 \) pointing in the same direction as \( \displaystyle \mathbf v = \left\langle 3, 0, -4 \right\rangle \).

\[ 5 \]
‖v‖.✓ Proved
\[ \left[\begin{matrix}6\\0\\-8\end{matrix}\right] \]
10·v/‖v‖.✓ Proved

Answer \( \left\langle 6, 0, -8 \right\rangle \)

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 5 and scales it by 10/5 to get the final vector. The steps are logically sound and the algebra is verified.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly calculates the magnitude of v as 5 and scales it by 10/5 to get the final vector. The steps are logically sound and the algebra is verified.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to state the method (scaling the vector by the ratio of desired magnitude to current magnitude) and provides no justification for the intermediate steps, making it impossible to verify the logic or learn the procedure.
gpt-oss:20b: pass 2026-10-05

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 2026-10-05 with SymPy 1.14.0.