Given a twice-differentiable vector-valued function $r\left(t\right)$, what is the definition of the principal unit normal vector $N\left(t\right)$?

Consider the given question,

A twice-differentiable vector-valued function is$r\left(t\right)$.

From the question,

Both $r\left(t\right),r\text{'}\text{'}\left(t\right)$exists.

It is clear that the principal unit normal vectors a unit vector with magnitude $1$and is orthogonal to the tangent vectors.

$T\left(t\right)$always points towards the direction in which the curve bends.

From the definition of the principal unit normal vector, if $r\left(t\right)$is a differentiable vector function $I\subseteq R$, then the principal unit normal vector at $r\left(t\right)$is denoted by $N\left(t\right)$is defined as,

$N\left(t\right)=\frac{T\text{'}\left(t\right)}{||T\text{'}\left(t\right)||}$

Where,$$\begin{gathered} T\text{'}\left(t\right)\ne 0, \\ T\left(t\right)=\frac{r\text{'}\left(t\right)}{||r\text{'}\left(t\right)||} \end{gathered}$$

Assume where $\alpha >0$.

From the given question,

Now the principal unit normal vector to is given below,