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Vectors in three dimensions

Problem 9.528 · medium

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

\[ 9 \]
‖v‖.✓ Proved
\[ \left[\begin{matrix}- \frac{16}{9}\\- \frac{16}{9}\\\frac{28}{9}\end{matrix}\right] \]
−4·v/‖v‖.✓ Proved

Answer \( \left\langle - \frac{16}{9}, - \frac{16}{9}, \frac{28}{9} \right\rangle \)

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.

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 9 and constructs the vector with magnitude 4 in the opposite direction by computing -4*v/||v||. The algebraic steps are verified and the final answer is correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly calculates the magnitude of v as 9 and constructs the vector with magnitude 4 in the opposite direction by computing -4*v/||v||. The algebraic steps are verified and the final answer is correct.
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly calculates the magnitude of v as 9 and constructs the vector with magnitude 4 in the opposite direction by computing -4*v/||v||. The algebraic steps are correct and the final answer matches the stated answer.
gpt-oss:20b: pass 2026-10-11

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.