Question

For each of the vector-valued functions, find the unit tangent vector.

$r\left(t\right)=(\cos \left(\alpha t\right),\sin \left(\alpha t\right))$

Short answer

$⟨-\sin \alpha t,\cos \alpha t⟩$

Step-by-step solution

Step1. Given Information

Step2. Continue

$\begin{array}{l}∥r^{\mathrm{\prime }}\left(t\right)∥=\parallel \sqrt{⟨-\alpha \sin \alpha t,\alpha \cos \alpha t⟩}\parallel \\ =\sqrt{⟨-\alpha \sin \alpha t{⟩}^2+(\alpha \cos \alpha t{)}^2}\\ =\sqrt{{\alpha }^2{\sin }^2\alpha t+{\alpha }^2{\cos }^2\alpha t}\\ =\sqrt{{\alpha }^2\left({\sin }^2\alpha t+{\cos }^2\alpha t\right)}\\ =\sqrt{{\alpha }^2}\\ =\alpha \\ \text{The unit tangent vector to}r\left(t\right)\text{is}\\ \begin{array}{l}T\left(t\right)=\frac{r^{\mathrm{\prime }}\left(t\right)}{∥r^{\prime \prime }\left(t\right)∥}\\ =\frac{⟨-\alpha \sin \alpha t,\alpha \cos \alpha t⟩}{\alpha }\\ =\frac{\alpha ⟨-\sin \alpha t,\cos \alpha t⟩}{\alpha }\\ =⟨-\sin \alpha t,\cos \alpha t⟩\\ \text{Thus the unit tangent vector to the given vector function is}\\ ⟨-\sin \alpha t,\cos \alpha t⟩\end{array}\end{array}$