Home›Calculus 3›Vectors in three dimensions›Problem 9.285

Vectors in three dimensions

Problem 9.285 · medium

Find the vector \( \displaystyle \mathbf u \) with \( \displaystyle \|\mathbf u\| = 7 \) pointing in the same direction as \( \displaystyle \mathbf v = \left\langle 1, 1, -1 \right\rangle \).

\[ \sqrt{3} \]
‖v‖.✓ Proved
\[ \left[\begin{matrix}\frac{7 \sqrt{3}}{3}\\\frac{7 \sqrt{3}}{3}\\- \frac{7 \sqrt{3}}{3}\end{matrix}\right] \]
7·v/‖v‖.✓ Proved

Answer \( \left\langle \frac{7 \sqrt{3}}{3}, \frac{7 \sqrt{3}}{3}, - \frac{7 \sqrt{3}}{3} \right\rangle \)

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 the vector to the desired length. The algebraic steps are verified and the final result is correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly computes the magnitude of v and scales the vector to the desired length. The algebraic steps are verified and the final result is correct.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution lacks any explanatory sentences describing the method (normalizing the vector and scaling by the desired magnitude). It only presents algebraic identities, failing to model the problem or justify the steps.
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.