---### Problem 1Let \(\vec{r}(t) = \langle \sin 3t, \cos 3t, \sqrt{7} t \rangle \).### Part (a)Find \(\vec{T}(t)\) and \(\vec{N}(t)\).### Part (b)Use your results from part (a) to find \(\vec{T}(0)\), \(\vec{N}(0)\), and \(\vec{B}(0)\).---In this problem, you are given a vector function \(\vec{r}(t)\) that defines a curve in 3-dimensional space. Your task is to find the unit tangent vector \(\vec{T}(t)\) and the principal normal vector \(\vec{N}(t)\).#### Explanation of Terms:- **\(\vec{r}(t)\)**: Position vector as a function of \(t\).- **\(\vec{T}(t)\)**: Unit tangent vector to the curve. It is the derivative of the position vector normalized.- **\(\vec{N}(t)\)**: Unit normal vector, which indicates the direction in which the curve is turning.Once you find \(\vec{T}(t)\) and \(\vec{N}(t)\), you will then find their values specifically at \(t = 0\). Finally, you'll find \(\vec{B}(0)\), which is the binormal vector at \(t = 0\).Remember, the binormal vector \(\vec{B}(t)\) is given by the cross product of the unit tangent vector \(\vec{T}(t)\) and the unit normal vector \(\vec{N}(t)\):\[\vec{B}(t) = \vec{T}(t) \times \vec{N}(t)\]Make sure to show all steps of your calculations clearly, explaining how each vector is derived from \(\vec{r}(t)\).---

Name: ---### Problem 1Let \(\vec{r}(t) = \langle \sin 3t, \cos 3t, \sqrt{7}
Uploaded: 2024-03-20T06:46:55.000Z
Channel: Bartleby
Description: ---### Problem 1Let \(\vec{r}(t) = \langle \sin 3t, \cos 3t, \sqrt{7} t \rangle \).### Part (a)Find \(\vec{T}(t)\) and \(\vec{N}(t)\).### Part (b)Use your results from part (a) to find \(\vec{T}(0)\), \(\vec{N}(0)\), and \(\vec{B}(0)\).---In this pr