Vectors in the plane
Problem 9.353 · easy
Find the angle \( \displaystyle \theta \in [0, 2\pi) \) that \( \displaystyle \mathbf u = \left\langle - \frac{5 \sqrt{3}}{2}, - \frac{5}{2} \right\rangle \) makes with the positive \( \displaystyle x \)-axis.
- \[ 5 \]‖u‖.✓ Proved
- \[ \left[\begin{matrix}- \frac{5 \sqrt{3}}{2}\\- \frac{5}{2}\end{matrix}\right] \]u = ‖u‖⟨cos θ, sin θ⟩ with θ = 7*pi/6, in the right quadrant.✓ Proved
Answer \( \theta = \frac{7 \pi}{6} \)
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 that the given angle corresponds to the vector components in the correct quadrant. The logic is sound and the answer is correct.
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 that the given angle corresponds to the vector components in the correct quadrant. The logic is sound and the answer is correct.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 the quadrant and reference angle were determined from the vector components.
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.