Vectors in the plane
Problem 9.351 · easy
For \( \displaystyle \mathbf a = \left\langle 1, 2 \right\rangle \) and \( \displaystyle \mathbf b = \left\langle -6, 4 \right\rangle \), find \( \displaystyle -2\mathbf a - 1\mathbf b \) and its magnitude.
- \[ \left[\begin{matrix}4\\-8\end{matrix}\right] \]Scale each vector, then add componentwise.✓ Proved
- \[ 4 \sqrt{5} \]Its length.✓ Proved
Answer \( \left\langle 4, -8 \right\rangle,\ \text{magnitude}\ 4 \sqrt{5} \)
✓ 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 | floating point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly computes the vector combination and its magnitude. The steps are algebraically sound and the final answer matches the stated result.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly computes the vector combination and its magnitude. The steps are algebraically sound and the final answer matches the stated result.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly computes the vector and its magnitude. The steps are algebraically sound and match 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_plane, checked 2026-10-07 with SymPy 1.14.0.