Vectors in the plane
Problem 9.516 · 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 = 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: fail (error) — The solution asserts that θ=π/3 without computing it from the components of u. It fails to show that cosθ=1/2 and sinθ=√3/2, which would justify θ=π/3. Thus the setup is incomplete and the conclusion is unsupported.qwen3.6:27b-mlx: fail (error) — The solution fails to calculate the angle; it merely verifies that the stated answer is consistent with the vector components. It does not demonstrate how to derive theta from the vector, which is the core of the problem.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-11 — The solution asserts that θ=π/3 without computing it from the components of u. It fails to show that cosθ=1/2 and sinθ=√3/2, which would justify θ=π/3. Thus the setup is incomplete and the conclusion is unsupported.qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to calculate the angle; it merely verifies that the stated answer is consistent with the vector components. It does not demonstrate how to derive theta from the vector, which is the core of the problem.qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to calculate the magnitude of the vector, which is 4, not 4 (wait, 2^2 + (2sqrt(3))^2 = 4 + 12 = 16, so magnitude is 4. The equation 4=4 is correct for magnitude squared? No, it just says 4=4. It's ambiguous and doesn't show the calculation). More importantly, the second line asserts the answer is pi/3 without showing the derivation (e.g., tan(theta) = sqrt(3)). It essentially states the answer rather than deriving it. Also, the first line '4=4' is not a standard way to present the magnitude calculation; usually one writes ||u|| = 4. The second line is a verification, not a derivation. A student needs to see how theta is found, e.g., cos(theta) = 2/4 = 1/2.gpt-oss:20b: pass 2026-10-11
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.