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Q. 9

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Found in: Page 879

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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

The definition of the principal unit normal vector $N\left(t\right)$ is given by,

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

See the step by step solution

## Step 1. Given information.

Consider the given question,

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

## Step 2. Write the definition.

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, $T\text{'}\left(t\right)\ne 0,\phantom{\rule{0ex}{0ex}}T\left(t\right)=\frac{r\text{'}\left(t\right)}{||r\text{'}\left(t\right)||}$

## Step 3. Consider an example.

Assume where $\alpha >0$.

## Step 4. Write the principal unit normal vector to rt.

From the given question,

Now the principal unit normal vector to is given below,