Vectors in the plane
Problem 9.278 · easy
Find the angle \( \displaystyle \theta \in [0, 2\pi) \) that \( \displaystyle \mathbf u = \left\langle \frac{5}{2}, \frac{5 \sqrt{3}}{2} \right\rangle \) makes with the positive \( \displaystyle x \)-axis.
- \[ 5 \]‖u‖.✓ Proved
- \[ \left[\begin{matrix}\frac{5}{2}\\\frac{5 \sqrt{3}}{2}\end{matrix}\right] \]u = ‖u‖⟨cos θ, sin θ⟩ with θ = pi/3, in the right quadrant.✓ Proved
Answer \( \theta = \frac{\pi}{3} \)
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: fail (misleading) — The solution merely verifies that the proposed answer satisfies the polar form equation, rather than deriving the angle from the vector components. This fails to demonstrate the method for finding the angle (e.g., using arctan) and does not address quadrant determination, which is crucial for a general solution.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (misleading) 2026-10-05 — The solution merely verifies that the proposed answer satisfies the polar form equation, rather than deriving the angle from the vector components. This fails to demonstrate the method for finding the angle (e.g., using arctan) and does not address quadrant determination, which is crucial for a general solution.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to explicitly calculate the angle using the vector components (e.g., via arctan) and instead asserts the answer by checking that the proposed angle yields the correct vector. This is a verification, not a derivation, and does not demonstrate how to find the angle from the given vector.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.