Q. 14 Unit tangent vectors: Find the u... [FREE SOLUTION]

Chapter 11: Q. 14 (page 901)

Unit tangent vectors: Find the unit tangent vector for the given function at the specified value of t.
$r (t) = 鉄� 3 \sin 2 t, 3 \cos 2 t 鉄�, t = \frac{蟺}{3}$

Short Answer

Expert verified

Ans: Thus the unit tangent vector to $鉄� 3 \sin 2 t, 3 \cos 2 t 鉄�$ at $t = \frac{蟺}{3}$ is $<\frac{- 1}{2}, \frac{- \sqrt{3}}{2}>$

Step by step solution

Step 1. Given information:

$r (t) = 鉄� 3 \sin 2 t, 3 \cos 2 t 鉄�, t = \frac{蟺}{3}$

Step 2. Simplifying the Unit tangent vectors :

Consider $r (t) = 鉄� 3 \sin 2 t, 3 \cos 2 t 鉄�$

First, we compute

$r^{'} (t) = \frac{d}{d t} 鉄� 3 \sin 2 t, 3 \cos 2 t 鉄� = 3 <\frac{d}{d t} (\sin 2 t), \frac{d}{d t} (\cos 2 t)> = 3 鉄� 2 \cos 2 t, - 2 \sin 2 t 鉄� = 6 鉄� \cos 2 t, - \sin 2 t 鉄� ||r^{'} (t)|| = 鈥� 6 鉄� \cos 2 t, - \sin 2 t 鉄� 鈥� = 6 \sqrt{(\cos 2 t)^{2} + (- \sin 2 t)^{2}} = 6 \sqrt{\cos^{2} 2 t + \sin^{2} 2 t}$

$= 6 s i n c e \sin^{2} 2 t + \cos^{2} 2 t = 1$

Step 3. Finding the Unit tangent vectors:

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

$T (t) = \frac{r^{'} (t)}{||r^{''} (t)||} = \frac{6 鉄� \cos 2 t, - \sin 2 t 鉄�}{6} = 鉄� \cos 2 t, - \sin 2 t 鉄�$

At $t = \frac{蟺}{3}$ , the unit tangent vector is

$T (\frac{蟺}{3}) = <\cos 2 (\frac{蟺}{3}), - \sin 2 (\frac{蟺}{3})> = <\cos 120 掳, - \sin 120 掳> = <\frac{- 1}{2}, \frac{- \sqrt{3}}{2}>$

Thus the unit tangent vector to $鉄� 3 \sin 2 t, 3 \cos 2 t 鉄�$ at $t = \frac{蟺}{3}$ is

$<\frac{- 1}{2}, \frac{- \sqrt{3}}{2}>$

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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.
$r (t) = 鉄� 3 \sin 2 t, 3 \cos 2 t 鉄�, t = \frac{蟺}{3}$

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�3sin2t,3cos2t鉄�,t=蟺3

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

Statistics

Theoretical and Mathematical Physics

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Calculus

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Study anywhere. Anytime. Across all devices.

Unit tangent vectors: Find the unit tangent vector for the given function at the specified value of t.
$r (t) = 鉄� 3 \sin 2 t, 3 \cos 2 t 鉄�, t = \frac{蟺}{3}$