Vectors in the plane
Problem 9.595 · easy
Find the unit vector in the direction of \( \displaystyle \mathbf v = \left\langle 3, -3 \right\rangle \).
- \[ 3 \sqrt{2} \]‖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: fail (error) — The solution does not correctly compute the unit vector. It incorrectly states that the norm is 3√2 without showing the calculation, and then merely repeats the same vector instead of dividing v by its norm to obtain the unit vector. The conclusion is therefore wrong.qwen3.6:27b-mlx: pass — The solution correctly computes the magnitude of the vector and divides the vector by this magnitude to find the unit vector. The algebraic steps are verified and the final answer is correct.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-13 — The solution does not correctly compute the unit vector. It incorrectly states that the norm is 3√2 without showing the calculation, and then merely repeats the same vector instead of dividing v by its norm to obtain the unit vector. The conclusion is therefore wrong.qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly computes the magnitude of the vector and divides the vector by this magnitude to find the unit vector. The algebraic steps are verified and the final answer is correct.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution fails to state the definition of a unit vector or the steps taken to normalize the vector. It presents the final answer as an unchecked equation without explaining how the magnitude was calculated or how the division was performed, which is insufficient for a calculus solution.
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 2026-10-13 with SymPy 1.14.0.