Vectors in three dimensions
Problem 9.446 · medium
Find the vector \( \displaystyle \mathbf u \) with \( \displaystyle \|\mathbf u\| = 4 \) pointing in the same 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: passqwen3.6:27b-mlx: pass — The solution correctly computes the magnitude of v as 9 and scales v by 4/9 to obtain the desired vector. The algebraic steps are verified.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly computes the magnitude of v as 9 and scales v by 4/9 to obtain the desired vector. The algebraic steps are verified.qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies the magnitude of v as 9 and computes the scaled vector 4*v/9, which matches the stated answer.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.