Vectors in the plane
Problem 9.359 · easy
Find the angle \( \displaystyle \theta \in [0, 2\pi) \) that \( \displaystyle \mathbf u = \left\langle \sqrt{3}, -1 \right\rangle \) makes with the positive \( \displaystyle x \)-axis.
- \[ 2 \]‖u‖.✓ Proved
- \[ \left[\begin{matrix}\sqrt{3}\\-1\end{matrix}\right] \]u = ‖u‖⟨cos θ, sin θ⟩ with θ = 11*pi/6, in the right quadrant.✓ Proved
Answer \( \theta = \frac{11 \pi}{6} \)
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 | atan2 |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the magnitude and verifies the angle satisfies the vector components and quadrant constraints.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the magnitude and verifies the angle satisfies the vector components and quadrant constraints.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution provides no derivation or reasoning for the angle, merely stating the final answer and verifying it post-hoc. It fails to explain how to determine the quadrant or use inverse trigonometric functions, which is the core of the problem.
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.