Calculus of vector-valued functions
Problem 9.418 · medium
Find the unit tangent vector \( \displaystyle \mathbf T \) of \( \displaystyle \mathbf r(t) = \left\langle \cos{\left(t \right)}, \sin{\left(t \right)}, t \right\rangle \) at \( \displaystyle t = 0 \).
- \[ \left[\begin{matrix}\frac{d}{d t} \cos{\left(t \right)}\\\frac{d}{d t} \sin{\left(t \right)}\\\frac{d}{d t} t\end{matrix}\right] = \left[\begin{matrix}- \sin{\left(t \right)}\\\cos{\left(t \right)}\\1\end{matrix}\right] \]r′(t).✓ Proved
- \[ \sqrt{2} \]‖r′(0)‖.✓ Proved
- \[ \left[\begin{matrix}0\\\frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{matrix}\right] \]T = r′/‖r′‖.✓ Proved
Answer \( \mathbf T(0) = \left\langle 0, \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right\rangle \)
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 |
| 3 | ✓ 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 | a numerical velocity, normalised |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly computes the derivative, evaluates its magnitude at t=0, and normalizes to find the unit tangent vector. The steps are logically sound and algebraically correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly computes the derivative, evaluates its magnitude at t=0, and normalizes to find the unit tangent vector. The steps are logically sound and algebraically correct.qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to explicitly calculate r'(0) before normalizing, jumping from the general derivative to the norm at t=0. While the final answer is correct, the logical gap (skipping the evaluation step) makes the derivation incomplete and potentially confusing for a student.gpt-oss:20b: pass 2026-10-08
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/vector_function_calculus, checked 2026-10-08 with SymPy 1.14.0.