Q. 21 Principal unit normal vectors: F... [FREE SOLUTION]

91影视

Chapter 11: Q. 21 (page 901)

Principal unit normal vectors: Find the principal unit normal vector for the given function at the specified value of t.
$r (t) = 鉄� 伪 \sin 尾 t, 伪 \cos 尾 t 鉄�$ , where $伪$ and $尾$ are positive, $t = 0$

Short Answer

Expert verified

Ans: The principal unit normal vector of $r (t) = 鉄� 伪 \sin 尾 t, 伪 \cos 尾 t 鉄�$ at $t = 0$ is $鉄� 0, - 1 鉄�$

Step by step solution

Step 1. Given information:

$r (t) = 鉄� 伪 \sin 尾 t, 伪 \cos 尾 t 鉄�$ , where $伪$ and $尾$ are positive, $t = 0$

Step 2. Simplifying the principal unit normal vector:

The principal unit normal vector of $r (t)$ denoted by $N (t)$ is given by

$N (t) = \frac{T^{'} (t)}{||T^{'} (t)||}$ where $T (t) = \frac{r^{'} (t)}{||r^{'} (t)||}$

$T (t)$ should be calculated first and then $N (t)$ is to be evaluated.

We start by comparing $r^{'} (t)$ first:

$r^{'} (t) = 鉄� 伪蟺 \cos 尾 t, - 伪尾 \sin 尾 t 鉄� ||r^{'} (t)|| = \sqrt{(伪尾 \cos 尾 t)^{2} + (- 伪尾 \sin 尾 t)^{2}} = \sqrt{伪^{2} 尾^{2} (\cos^{2} 尾 t + \sin^{2} 尾 t)} = 伪尾$

The unit tangent vector

$T (t) = \frac{r^{'} (t)}{||r^{'} (t)||} = \frac{鉄� 伪尾 \cos 尾 t, - 伪尾 \sin 尾 t 鉄�}{伪尾} = 鉄� \cos 尾 t, - \sin 尾 t 鉄�$

To find $N (t)$ we divide $T^{'} (t)$ by its magnitude.

$T^{'} (t) = 鉄� - 尾 \sin 尾 t, - 尾 \cos 尾 t 鉄� ||T^{'} (t)|| = \sqrt{(- 尾 \sin 尾 t)^{2} + (- 尾 \cos 尾 t)^{2}} = \sqrt{尾^{2} (\sin^{2} 尾 t + \cos^{2} 尾 t)} = 尾$

Step 3. Finding the principal unit normal vector:

The principal unit normal vector

$N (t) = \frac{T^{'} (t)}{||T^{'} (t)||} = \frac{鉄� - 尾 \sin 尾 t, - 尾 \cos 尾 t 鉄�}{尾} = 鉄� - \sin 尾 t, - \cos 尾 t 鉄�$

At $t = 0, N (t) = N (0) = 鉄� - \sin 0, - \cos 0 鉄� = 鉄� 0, - 1 鉄�$

Thus the principle unit normal vector of $r (t)$ at $t = 0$ is $鉄� 0, - 1 鉄�$

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

Principal unit normal vectors: Find the principal unit normal vector for the given function at the specified value of t.
$r (t) = 鉄� 伪 \sin 尾 t, 伪 \cos 尾 t 鉄�$ , where $伪$ and $尾$ are positive, $t = 0$

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Simplifying the principal unit normal vector:

Step 3. Finding the principal unit normal vector:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Theoretical and Mathematical Physics

Geometry

Applied Mathematics

Pure Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

91影视

Principal unit normal vectors: Find the principal unit normal vector for the given function at the specified value of t.r(t)=鉄�伪sin尾t,伪cos尾t鉄�, where 伪 and 尾 are positive,t=0

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Simplifying the principal unit normal vector:

Step 3. Finding the principal unit normal vector:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Theoretical and Mathematical Physics

Geometry

Applied Mathematics

Pure Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Principal unit normal vectors: Find the principal unit normal vector for the given function at the specified value of t.
$r (t) = 鉄� 伪 \sin 尾 t, 伪 \cos 尾 t 鉄�$ , where $伪$ and $尾$ are positive, $t = 0$