Question

Unit tangent vectors: Find the unit tangent vector for the given function at the specified value of t.

$\mathbf{r}\left(t\right)=<\alpha \sin \beta t,\alpha \cos \beta t>,t=0$

Short answer

Ans: The unit tangent vector to$<\alpha \sin \beta t,\alpha \cos \beta t>$at $t=0$is $⟨1,0⟩$

Step-by-step solution

Step 1. Given information:

$\mathbf{r}\left(t\right)=<\alpha \sin \beta t,\alpha \cos \beta t>,t=0$

Step 2. Simplifying the Unit tangent vectors :

Consider $r\left(t\right)=⟨\alpha \sin \beta t,\alpha \cos \beta t⟩$

First, we compute

Step 3. Finding the Unit tangent vectors: 

The unit tangent vector to $r\left(t\right)$is

$$\begin{gathered} T\left(t\right)=\frac{r^{\text{'}}\left(t\right)}{||r^{\text{'}\text{'}}\left(t\right)||} \\ =\frac{⟨\alpha \beta \cos \beta t,-\alpha \beta \sin \beta t⟩}{\alpha \beta } \\ =\frac{\alpha \beta ⟨\cos \beta t,-\sin \beta t⟩}{\alpha \beta } \\ =⟨\cos \beta t,-\sin \beta t⟩ \end{gathered}$$

At $t=0$, the unit tangent vector to $r\left(t\right)$is

$$\begin{gathered} T\left(0\right)=⟨\cos \beta \left(0\right),-\sin \beta \left(0\right)⟩ \\ =⟨\cos 0,-\sin 0⟩ \\ =⟨1,0⟩ \end{gathered}$$

Thus the unit tangent vector to $⟨\alpha \sin \beta t,\alpha \cos \beta t⟩$at $t=0$is $⟨1,0⟩$