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.
$r (t) = (3 \sin (t), 5 \cos (t), 4 \sin (t)), t = π$

The unit tangent vector and the principle unit normal vector are

$T (π) = (\frac{- 3}{5}, 0, \frac{- 4}{5}) N (π) = (0, 1, 0)$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information

Given $r (t) = (3 \sin (t), 5 \cos (t), 4 \sin (t)), t = π$

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

$r (t) = (3 \sin (t), 5 \cos (t), 4 \sin (t)) r' (t) = (3 \cos (t), - 5 \sin (t), 4 \cos (t)) ∥ r' (t) ∥ = \sqrt{(3^{2} \cos^{2} (t) + 5^{2} \sin^{2} (t) + 4^{2} \cos^{2} (t))} = \sqrt{25 \sin^{2} (t) + 25 \cos^{2} (t)} = 5 P a r t (a) T (t) = \frac{r' (t)}{∥ r' (t) ∥} = \frac{(3 \cos (t), - 5 \sin (t), 4 \cos (t))}{5} T (π) = \frac{(3 \cos (π), - 5 \sin (π), 4 \cos (π))}{5} = (\frac{- 3, 0, - 4}{5}) = (\frac{- 3}{5}, 0, \frac{- 4}{5}) T' (t) = \frac{1}{5} (3 \sin (t), - 5 \cos (t), 4 \sin (t)) ∥ T' (t) ∥ = \frac{1}{5} \sqrt{(3^{2} \sin^{2} (t) + (- 5^{2} \cos^{2} (t)) + 4^{2} \sin^{2} (t))} = \frac{1}{5} \sqrt{25 \sin^{2} (t) + 25 \cos^{2} (t)} = 1 P a r t (b) N (t) = \frac{T' (t)}{∥ T' (t) ∥} = \frac{\frac{1}{5} (3 \sin (t), - 5 \cos (t), 4 \sin (t))}{1} N (t) = \frac{1}{5} (3 \sin (t), - 5 \cos (t), 4 \sin (t)) A t t = π N (π) = \frac{1}{5} (3 \sin (π), - 5 \cos (π), 4 \sin (π)) = \frac{1}{5} (0, 5, 0) = (0, 1, 0)$

Step 3. The solution

The unit tangent vector and principle unit normal vector are

$T (π) = (\frac{- 3}{5}, 0, \frac{- 4}{5}) N (π) = (0, 1, 0)$

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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.
$r (t) = (3 \sin (t), 5 \cos (t), 4 \sin (t)), t = π$

Step by Step Solution

Step 1. Given information

Step 2. Parta. Find unit 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.
$r (t) = (3 \sin (t), 5 \cos (t), 4 \sin (t)), t = π$