Calculus of vector-valued functions
Problem 9.252 · medium
Find the unit tangent vector \( \displaystyle \mathbf T \) of \( \displaystyle \mathbf r(t) = \left\langle t, t^{2} - 1, 2 t \right\rangle \) at \( \displaystyle t = 0 \).
- \[ \left[\begin{matrix}\frac{d}{d t} t\\\frac{d}{d t} \left(t^{2} - 1\right)\\\frac{d}{d t} 2 t\end{matrix}\right] = \left[\begin{matrix}1\\2 t\\2\end{matrix}\right] \]r′(t).✓ Proved
- \[ \sqrt{5} \]‖r′(0)‖.✓ Proved
- \[ \left[\begin{matrix}\frac{\sqrt{5}}{5}\\0\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] \]T = r′/‖r′‖.✓ Proved
Answer \( \mathbf T(0) = \left\langle \frac{\sqrt{5}}{5}, 0, \frac{2 \sqrt{5}}{5} \right\rangle \)
✓ 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 |
| 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 the vector to find the unit tangent vector. The steps are logically sound and the final answer is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly computes the derivative, evaluates its magnitude at t=0, and normalizes the vector to find the unit tangent vector. The steps are logically sound and the final answer is correct.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly computes the derivative, evaluates its magnitude at t=0, and normalizes the vector to find the unit tangent vector. The steps are logically sound and the final answer is correct.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.