Vectors in the plane
Problem 9.279 · easy
Find the unit vector in the direction of \( \displaystyle \mathbf v = \left\langle 4, -4 \right\rangle \).
- \[ 4 \sqrt{2} \]‖v‖.✓ Proved
- \[ \left[\begin{matrix}\frac{\sqrt{2}}{2}\\\frac{\left(-1\right) \sqrt{2}}{2}\end{matrix}\right] = \left[\begin{matrix}\frac{\sqrt{2}}{2}\\- \frac{\sqrt{2}}{2}\end{matrix}\right] \]v/‖v‖.✓ Proved
Answer \( \left\langle \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2} \right\rangle \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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 computes the magnitude and the unit 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 computes the magnitude and the unit vector. The lack of explanatory text is acceptable given the simplicity of the problem.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution lacks any explanatory sentences describing the method (dividing by magnitude) or the calculation of the magnitude. It only provides unchecked algebraic identities, failing to demonstrate the logical steps required to solve the problem.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.