Home›Calculus 3›Vectors in the plane›Problem 9.596

Vectors in the plane

Problem 9.596 · easy

Find the angle \( \displaystyle \theta \in [0, 2\pi) \) that \( \displaystyle \mathbf u = \left\langle \frac{1}{2}, - \frac{\sqrt{3}}{2} \right\rangle \) makes with the positive \( \displaystyle x \)-axis.

\[ 1 \]
‖u‖.✓ Proved
\[ \left[\begin{matrix}1 \cdot \frac{1}{2}\\\frac{\left(-1\right) \sqrt{3}}{2}\end{matrix}\right] = \left[\begin{matrix}\frac{1}{2}\\- \frac{\sqrt{3}}{2}\end{matrix}\right] \]
u = ‖u‖⟨cos θ, sin θ⟩ with θ = 5*pi/3, in the right quadrant.✓ Proved

Answer \( \theta = \frac{5 \pi}{3} \)

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.

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 angle by verifying that the given vector matches the polar form with the specified angle, confirming both magnitude and quadrant.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-13
qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the angle by verifying that the given vector matches the polar form with the specified angle, confirming both magnitude and quadrant.
gpt-oss:20b: pass 2026-10-13
qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the magnitude and verifies the angle using the standard polar form definition, confirming the quadrant and value.

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 2026-10-13 with SymPy 1.14.0.