Q. 17 Unit tangent vectors: Find the u... [FREE SOLUTION]

Chapter 11: Q. 17 (page 901)

Unit tangent vectors: Find the unit tangent vector for the given function at the specified value of t.
$r (t) = 鉄� t, 2 t \sin t, 2 t \cos t 鉄�, t = 蟺$

Short Answer

Expert verified

Ans: Thus the unit tangent vector to $鉄� t, 2 t \sin t, 2 t \cos t 鉄�$ at $t = 蟺$ islocalid="1649674946140" $<\frac{1}{\sqrt{4 蟺^{2} + 5}}, \frac{- 2 蟺}{\sqrt{4 蟺^{2} + 5}}, \frac{- 2}{\sqrt{4 蟺^{2} + 5}}>$

Step by step solution

Step 1. Given information:

$r (t) = 鉄� t, 2 t \sin t, 2 t \cos t 鉄�, t = 蟺$

Step 2. Simplifying the Unit tangent vectors :

Consider $r (t) = 鉄� t, 2 t \sin t, 2 t \cos t 鉄�$

First, we compute $r^{'} (t)$

$r^{'} (t) = \frac{d}{d t} 鉄� t, 2 t \sin t, 2 t \cos t 鉄� = 鉄� 1, 2 (t \cos t + \sin t), 2 (- t \sin t + \cos t) 鉄� ||r^{'} (t)|| = \sqrt{(1)^{2} + (2 (t \cos t + \sin t))^{2} + (2 (- t \sin t + \cos t))^{2}} = \sqrt{1 + 4 (t^{2} \cos^{2} t + \sin^{2} t + 2 t \sin t \cos t + t^{2} \sin^{2} t + \cos^{2} t - 2 t \sin t \cos t)} = \sqrt{1 + 4 (t^{2} + 1)} = \sqrt{4 t^{2} + 5}$

Step 3. Finding the Unit tangent vectors:

The unit tangent vector to $r (t)$ is

$T (t) = \frac{r^{'} (t)}{||r^{'} (t)||} = \frac{鉄� 1, 2 (t \cos t + \sin t), 2 (- t \sin t + \cos t) 鉄�}{\sqrt{4 t^{2} + 5}}$

At $t = 蟺$ , the unit tangent vector to $r (t)$ is

localid="1649674786888" $T (蟺) = \frac{鉄� 1, 2 (蟺 \cos 蟺 + \sin 蟺), 2 (- 蟺 \sin 蟺 + \cos 蟺) 鉄�}{\sqrt{4 (蟺)^{2} + 5}} = \frac{鉄� 1, - 2 蟺, - 2 鉄�}{\sqrt{4 蟺^{2} + 5}}$

Thus the unit tangent vector to $鉄� t, 2 t \sin t, 2 t \cos t 鉄�$ at $t = 蟺$ is

$<\frac{1}{\sqrt{4 蟺^{2} + 5}}, \frac{- 2 蟺}{\sqrt{4 蟺^{2} + 5}}, \frac{- 2}{\sqrt{4 蟺^{2} + 5}}>$

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91影视

Unit tangent vectors: Find the unit tangent vector for the given function at the specified value of t.
$r (t) = 鉄� t, 2 t \sin t, 2 t \cos t 鉄�, t = 蟺$

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Simplifying the Unit tangent vectors :

Step 3. Finding the Unit tangent vectors:

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91影视

Unit tangent vectors: Find the unit tangent vector for the given function at the specified value of t.r(t)=鉄�t,2tsint,2tcost鉄�,t=蟺

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Simplifying the Unit tangent vectors :

Step 3. Finding the Unit tangent vectors:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Theoretical and Mathematical Physics

Calculus

Decision Maths

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Unit tangent vectors: Find the unit tangent vector for the given function at the specified value of t.
$r (t) = 鉄� t, 2 t \sin t, 2 t \cos t 鉄�, t = 蟺$