Vectors in the plane
Problem 9.274 · easy
Find the unit vector in the direction of \( \displaystyle \mathbf v = \left\langle -2, 2 \right\rangle \).
- \[ 2 \sqrt{2} \]‖v‖.✓ Proved
- \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{matrix}\right] \]v/‖v‖.✓ Proved
Answer \( \left\langle - \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right\rangle \)
✓ 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 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 calculates the magnitude and normalizes the vector. The algebraic steps are verified and the final result is correct.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly computes the magnitude of the vector and divides the vector by this magnitude to obtain the unit vector. The algebraic steps are correct and the final answer matches the stated answer.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.