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Chapter 11: Q. 7 (page 879)

Given a differentiable vector-valued function r(t), what is the relationship between r't0and Tt0at a pointt0in the domain of r(t)?

Short Answer

Expert verified

The relationship betweenr't0,Tt0 isTt=r'tr't.

Step by step solution

01

Step 1. Given information.

Consider the given question,

A differentiable vector-valued function isrt.

02

Step 2. Write the relationship between r't0,Tt0.

Assume t0be any point in the domain of rt. The unit tangent vector of rtis denoted by Tt.

According to the definition of unit tangent vector function on some interval I鈯�Rsuch that r't鈮�0 on I.

Then the unit tangent function can be defined as,

Tt=r'tr't

Where,r't0鈮�0

03

Step 3. Consider the definition.

Consider the definition it is clear that,

Both Tt0,r't0have same direction.

Also, role="math" localid="1649564983175" r't0,Tt0may differ in lengths unless r't0=1, as Tt0always have the magnitude of 1unit because it is a unit vector of rt.

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Most popular questions from this chapter

In Exercises 19鈥�21 sketch the graph of a vector-valued function r(t)=x(t),y(t) with the specified properties. Be sure to indicate the direction of increasing values oft.

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