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Vectors in three dimensions

Problem 9.521 · medium

Find the vector \( \displaystyle \mathbf u \) with \( \displaystyle \|\mathbf u\| = 2 \) pointing in the same direction as \( \displaystyle \mathbf v = \left\langle 4, -4, 7 \right\rangle \).

\[ 9 \]
‖v‖.✓ Proved
\[ \left[\begin{matrix}\frac{8}{9}\\- \frac{8}{9}\\\frac{14}{9}\end{matrix}\right] \]
2·v/‖v‖.✓ Proved

Answer \( \left\langle \frac{8}{9}, - \frac{8}{9}, \frac{14}{9} \right\rangle \)

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 as 9 and scales v by 2/9 to obtain the desired vector. The algebraic steps are verified and the logic is sound.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly computes the magnitude of v as 9 and scales v by 2/9 to obtain the desired vector. The algebraic steps are verified and the logic is sound.
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution lacks the necessary sentence explaining the method (scaling the unit vector of v by the desired magnitude). Without this, the steps are just unexplained calculations.

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.