Chapter 11: Q. 14 (page 901)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information:

$r (t) = ⟨ 3 \sin 2 t, 3 \cos 2 t ⟩, t = \frac{π}{3}$

Step 2. Simplifying the Unit tangent vectors :

Consider $r (t) = ⟨ 3 \sin 2 t, 3 \cos 2 t ⟩$

First, we compute

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

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

Step 3. Finding the Unit tangent vectors:

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

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

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

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

Thus the unit tangent vector to $⟨ 3 \sin 2 t, 3 \cos 2 t ⟩$ at $t = \frac{π}{3}$ is

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

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

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

Geometry

Applied Mathematics

Calculus

Pure Maths

Probability and Statistics

Theoretical and Mathematical Physics

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)=⟨3sin2t,3cos2t⟩,t=π3

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

Geometry

Applied Mathematics

Calculus

Pure Maths

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

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