Short Answer

For each of the vector-valued functions in Exercises , find the unit tangent vector and the principal unit normal vector at the specified value of t.
$r (t) = (\cos (α t), \sin (α t)), w h e r e α > 0, t = π$

The unit tangent vector and principle unit normal vector are

$T (π) = (- \sin (α π), \cos (α π)) N (π) = (- \cos (α π), - \sin (α π))$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information

Given $r (t) = (\cos (α t), \sin (α t)), t = π$

Step 2..Parta. Find tangent vector Partb. Find principle unit normal vector

$r (t) = (\cos (α t), \sin (α t)) r' (t) = (- α \sin (α t), α \cos (α t)) ∥ r' (t) ∥ = ∥ \{- α \sin (α t), α \cos (α t)\} ∥ = \sqrt{{(- α \sin (α t))}^{2} + {(α \cos (α t))}^{2}} = \sqrt{α^{2} (\sin^{2} (α t) + \cos^{2} (α t))} = α P a r t (a) T (t) = \frac{r' (t)}{∥ r' (t) ∥} = \frac{(- α \sin (α t), α \cos (α t))}{α} = (- \sin (α t), \cos (α t)) A t t = π, T (π) = (- \sin (α π), \cos (α π)) T' (t) = (- α \cos (α t), - α \sin (α t)) ∥ T' (t) ∥ = \sqrt{α^{2} (\cos^{2} (α t) + \sin^{2} (α t))} = α P a r t (b) N (t) = \frac{T' (t)}{∥ T' (t) ∥} = \frac{(- α \cos (α t), - α \sin (α t))}{α} = (- \cos (α t), - \sin (α t)) A t t = π N (π) = (- \cos (απ), - \sin (απ))$

Step 3. The solution

The unit tangent vector and principle unit normal vector is

$T (π) = (- \sin (α π), \cos (α π)) N (π) = (- \cos (α π), - \sin (α π))$

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Short Answer

For each of the vector-valued functions in Exercises , find the unit tangent vector and the principal unit normal vector at the specified value of t.
$r (t) = (\cos (α t), \sin (α t)), w h e r e α > 0, t = π$

Step by Step Solution

Step 1. Given information

Step 2..Parta. Find tangent vector Partb. Find principle unit normal vector

Step 3. The solution

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For each of the vector-valued functions in Exercises , find the unit tangent vector and the principal unit normal vector at the specified value of t. rt=cosαt, sinαt, where α>0, t=π

Step by Step Solution

Step 1. Given information

Step 2..Parta. Find tangent vector Partb. Find principle unit normal vector

Step 3. The solution

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For each of the vector-valued functions in Exercises , find the unit tangent vector and the principal unit normal vector at the specified value of t.
$r (t) = (\cos (α t), \sin (α t)), w h e r e α > 0, t = π$