$(15$ pts$.)3.$ Consider the curve, $C,$ deteremined by the vector valued function vec$(F)(t)=($:$sin(2t),t,cos(2t)$:$)$ on the interval $0,2$ and the point $P=(0,,1)$ on the curve.

$($i$)$ Sketch the curve.

$($ii$)$ Determine the length of the curve.

$($iii$)$ Determine the curvature of the curve $C$ at the point $P$

$($iv$)$ Find the unit tangent vector, vec$(T)$ and unit normal vector vec$(N)$ at $P.$

$($v$)$ Find the center and radius of the circle of curvature for the curve $C$ at the point $P.$