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Vectors in three dimensions

Problem 9.445 · medium

Find the vector \( \displaystyle \mathbf u \) with \( \displaystyle \|\mathbf u\| = 5 \) pointing in the same direction as \( \displaystyle \mathbf v = \left\langle -3, 0, -4 \right\rangle \).

\[ 5 \]
‖v‖.✓ Proved
\[ \left[\begin{matrix}-3\\0\\-4\end{matrix}\right] \]
5·v/‖v‖.✓ Proved

Answer \( \left\langle -3, 0, -4 \right\rangle \)

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: fail (misleading) — The solution fails to explain the method of normalizing the vector and scaling it by the desired magnitude. It merely asserts the result via tautological equations, teaching the student nothing about how to find a vector of a specific length in a given direction.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: fail (misleading) 2026-10-09 — The solution fails to explain the method of normalizing the vector and scaling it by the desired magnitude. It merely asserts the result via tautological equations, teaching the student nothing about how to find a vector of a specific length in a given direction.
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to explain the method of normalizing the vector v and scaling it by the desired magnitude. It presents tautological equations without deriving the result, making it impossible to follow the logic.
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.