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Vectors in the plane

Problem 9.356 · easy

Find the angle \( \displaystyle \theta \in [0, 2\pi) \) that \( \displaystyle \mathbf u = \left\langle \frac{5 \sqrt{2}}{2}, \frac{5 \sqrt{2}}{2} \right\rangle \) makes with the positive \( \displaystyle x \)-axis.

\[ 5 \]
‖u‖.✓ Proved
\[ \left[\begin{matrix}\frac{5 \sqrt{2}}{2}\\\frac{5 \sqrt{2}}{2}\end{matrix}\right] \]
u = ‖u‖⟨cos θ, sin θ⟩ with θ = pi/4, in the right quadrant.✓ Proved

Answer \( \theta = \frac{\pi}{4} \)

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 the angle using the standard polar form, confirming the vector lies in the first quadrant with equal components.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the magnitude and verifies the angle using the standard polar form, confirming the vector lies in the first quadrant with equal components.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to actually solve for theta; it merely verifies that the given answer is consistent with the vector components. It does not demonstrate the derivation (e.g., using arctan) required to find the angle.

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.