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Chapter 11: Q. 15 (page 901)

Unit tangent vectors: Find the unit tangent vector for the given function at the specified value of t.

r(t)=<伪sin尾t,伪cos尾t>,t=0

Short Answer

Expert verified

Ans: The unit tangent vector to<伪sin尾t,伪cos尾t>at t=0is 鉄�1,0鉄�

Step by step solution

01

Step 1. Given information:

r(t)=<伪sin尾t,伪cos尾t>,t=0

02

Step 2. Simplifying the Unit tangent vectors :

Consider r(t)=鉄�伪sin尾t,伪cos尾t鉄�

First, we compute

r'(t)=ddt鉄�伪sin尾t,伪cos尾t鉄�=ddt(伪cos尾t),ddt(伪cos尾t)=鉄�伪cos尾t,-伪尾sin尾t鉄�r'(t)=鈥�鉄�伪尾cos尾t,-伪尾sin尾t鉄�鈥�=(伪尾cos尾t)2+(-伪尾sin尾t)2=伪尾cos2尾t+sin2尾t=伪尾

03

Step 3. Finding the Unit tangent vectors: 

The unit tangent vector to r(t)is

T(t)=r'(t)r''(t)=鉄�伪尾cos尾t,-伪尾sin尾t鉄�伪尾=伪尾鉄�cos尾t,-sin尾t鉄�伪尾=鉄�cos尾t,-sin尾t鉄�

At t=0, the unit tangent vector to r(t)is

T(0)=鉄�cos尾(0),-sin尾(0)鉄�=鉄�cos0,-sin0鉄�=鉄�1,0鉄�

Thus the unit tangent vector to 鉄�伪sin尾t,伪cos尾t鉄�at t=0is 鉄�1,0鉄�

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