Vectors in the plane
Problem 9.511 · easy
Find the angle \( \displaystyle \theta \in [0, 2\pi) \) that \( \displaystyle \mathbf u = \left\langle -2, - 2 \sqrt{3} \right\rangle \) makes with the positive \( \displaystyle x \)-axis.
- \[ 4 \]‖u‖.✓ Proved
- \[ \left[\begin{matrix}-2\\- 2 \sqrt{3}\end{matrix}\right] \]u = ‖u‖⟨cos θ, sin θ⟩ with θ = 4*pi/3, in the right quadrant.✓ Proved
Answer \( \theta = \frac{4 \pi}{3} \)
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: fail (error) — The solution skips the essential steps of computing the magnitude and determining the correct angle from the components. It merely states the result without justification, so the setup and conclusion are incomplete and incorrect.qwen3.6:27b-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: fail (error) 2026-10-11 — The solution skips the essential steps of computing the magnitude and determining the correct angle from the components. It merely states the result without justification, so the setup and conclusion are incomplete and incorrect.qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly identifies the magnitude and verifies the angle satisfies the vector components and quadrant constraints.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly identifies the magnitude and verifies that the given angle corresponds to the vector components in the correct quadrant.
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 2026-10-11 with SymPy 1.14.0.