Vectors in the plane
Problem 9.280 · easy
Find the unit vector in the direction of \( \displaystyle \mathbf v = \left\langle -6, -8 \right\rangle \).
- \[ 10 \]‖v‖.✓ Proved
- \[ \left[\begin{matrix}- \frac{3}{5}\\- \frac{4}{5}\end{matrix}\right] \]v/‖v‖.✓ Proved
Answer \( \left\langle - \frac{3}{5}, - \frac{4}{5} \right\rangle \)
✓ 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 1 and parallel to v, same direction |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly calculates the magnitude and normalizes the vector. The lack of explanatory text is acceptable given the simplicity of the problem.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly calculates the magnitude and normalizes the vector. The lack of explanatory text is acceptable given the simplicity of the problem.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly calculates the magnitude and the unit vector. Although the steps are presented as tautological equations without explicit derivation sentences, the mathematical content is correct and the final answer matches the problem statement.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_plane, checked 2026-10-05 with SymPy 1.14.0.