Chapter 12.4, Problem 11E

### Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

Chapter
Section

### Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem
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# Finding a Tangent Line In Exercises 9-14, find the unit tangent vector T( t) and a set of parametric equations for the tangent line to the space curve at point P. r ( t ) = 〈 2 cos t , 2 sin t , 4 〉 , P ( 2 , 2 , 4 )

To determine

To calculate: The unit tangent vector and a set of parametric equations for the tangent line to the space curve r(t)=2cost,2sint,4 at point P (2,2,4).

Explanation

Given:

The curve: r(t)=2cost,2sint,4

And the point P: (2,2,4)

Formula Used:

Let C be a smooth curve which makes the representation by r on an open interval I. The unit tangent vector at t is defined as:

T(t)=r(t)r(t),  r(t)0

A line L parallel to v=a,b,c and passing through the point P(x1,y1,z1) is shown by the parametric equation:

x=x1+aty=y1+btz=z1+ct

Calculation:

Consider the specified curve:

r(t)=2cost,2sint,4

Differentiate r(t) as follows:

r(t)=ddt2cost,2sint,4=2sint,2cost,0

This gives the unit tangent vector as follows:

T(t)=r(t)r(t)=1(2sint)2+(2cost)2+(0)22sint,2cost,0

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