Q. 31 For each of the vector-valued fu... [FREE SOLUTION]

91影视

Chapter 11: Q. 31 (page 880)

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.
$r (t) = (\cos (伪 t), \sin (伪 t)), w h e r e 伪 > 0, t = 蟺$

Short Answer

Expert verified

The unit tangent vector and principle unit normal vector are

$T (蟺) = (- \sin (伪蟺), \cos (伪蟺)) N (蟺) = (- \cos (伪蟺), - \sin (伪蟺))$

Step by step solution

Step 1. Given information

Given $r (t) = (\cos (伪 t), \sin (伪 t)), t = 蟺$

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

$r (t) = (\cos (伪 t), \sin (伪 t)) r' (t) = (- 伪 \sin (伪 t), 伪 \cos (伪 t)) 鈭� r' (t) 鈭� = 鈭� \{- 伪 \sin (伪 t), 伪 \cos (伪 t)\} 鈭� = \sqrt{{(- 伪 \sin (伪 t))}^{2} + {(伪 \cos (伪 t))}^{2}} = \sqrt{伪^{2} (\sin^{2} (伪 t) + \cos^{2} (伪 t))} = 伪 P a r t (a) T (t) = \frac{r' (t)}{鈭� r' (t) 鈭�} = \frac{(- 伪 \sin (伪 t), 伪 \cos (伪 t))}{伪} = (- \sin (伪 t), \cos (伪 t)) A t t = 蟺, T (蟺) = (- \sin (伪蟺), \cos (伪蟺)) T' (t) = (- 伪 \cos (伪 t), - 伪 \sin (伪 t)) 鈭� T' (t) 鈭� = \sqrt{伪^{2} (\cos^{2} (伪 t) + \sin^{2} (伪 t))} = 伪 P a r t (b) N (t) = \frac{T' (t)}{鈭� T' (t) 鈭�} = \frac{(- 伪 \cos (伪 t), - 伪 \sin (伪 t))}{伪} = (- \cos (伪 t), - \sin (伪 t)) A t t = 蟺 N (蟺) = (- \cos (伪蟺), - \sin (伪蟺))$

Step 3. The solution

The unit tangent vector and principle unit normal vector is

$T (蟺) = (- \sin (伪蟺), \cos (伪蟺)) N (蟺) = (- \cos (伪蟺), - \sin (伪蟺))$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

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.
$r (t) = (\cos (伪 t), \sin (伪 t)), w h e r e 伪 > 0, t = 蟺$

Short Answer

Step by step solution

Step 1. Given information

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

Step 3. The solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Calculus

Mechanics Maths

Pure Maths

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

91影视

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=cos伪t,sin伪t,where伪>0,t=蟺

Short Answer

Step by step solution

Step 1. Given information

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

Step 3. The solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Calculus

Mechanics Maths

Pure Maths

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

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.
$r (t) = (\cos (伪 t), \sin (伪 t)), w h e r e 伪 > 0, t = 蟺$