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Vectors in the plane

Problem 9.276 · 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 = pi/6, in the right quadrant.✓ Proved

Answer \( \theta = \frac{\pi}{6} \)

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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the magnitude and verifies that the given angle satisfies the polar coordinate definition for the vector in the first quadrant.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the magnitude and verifies that the given angle satisfies the polar coordinate definition for the vector in the first quadrant.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to show the derivation of the angle, merely stating the result and verifying it. It does not demonstrate the use of trigonometric ratios (e.g., tan(theta) = 1/sqrt(3)) required to solve the problem.
gpt-oss:20b: fail (misleading) 2026-10-05 — The solution states that θ=π/6 and that u=‖u‖⟨cosθ,sinθ⟩ without showing the computation that cosθ=√3/2 and sinθ=1/2. The justification is missing, so a student could be misled into thinking the result is proven when it is not.

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.