Vectors in three dimensions
Problem 9.370 · medium
Find the vector \( \displaystyle \mathbf u \) with \( \displaystyle \|\mathbf u\| = 8 \) pointing in the opposite direction as \( \displaystyle \mathbf v = \left\langle -1, 4, -8 \right\rangle \).
- \[ 9 \]‖v‖.✓ Proved
- \[ \left[\begin{matrix}\frac{8}{9}\\- \frac{32}{9}\\\frac{64}{9}\end{matrix}\right] \]−8·v/‖v‖.✓ Proved
Answer \( \left\langle \frac{8}{9}, - \frac{32}{9}, \frac{64}{9} \right\rangle \)
✓ 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 calculates the norm of v as 9 and computes the vector u as -8*v/||v||, which yields the correct result. The steps are logically sound and algebraically verified.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly calculates the norm of v as 9 and computes the vector u as -8*v/||v||, which yields the correct result. The steps are logically sound and algebraically verified.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly calculates the magnitude of v as 9 and computes the vector in the opposite direction with magnitude 8 as -8v/||v||. The final vector matches the stated answer.gpt-oss:20b: pass 2026-10-07
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 2026-10-07 with SymPy 1.14.0.