Chapter 12.4, Problem 20E

### 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 the Principal Unit Normal VectorIn Exercises 15–20, find the principal unit normal vector to the curve at the specified value of the parameter. r ( t ) = cos 3 t i + 2 sin 3 t j + k ,       t = π

To determine

To calculate: The unit normal vector to curve, r(t)=cos3ti+2sin3tj+k at the specified value of the parameter, t=π.

Explanation

Given:

The provided curve is: r(t)=cos3ti+2sin3tj+k.

And the specified value of the parameter is: t=π.

Formula used:

Consider that C be a smooth curve represented by r over an open interval I.

Then, the unit tangent vector at t is shown below as:

T(t)=r(t)r(t),  r(t)0

And the unit normal vector at t is:

N(t)=T(t)T(t),  T(t)0

Calculation:

The provided curve is,

r(t)=cos3ti+2sin3tj+k

Now, the derivative of curve r(t) is:

r(t)=(ddtcos3t)i+(ddt2sin3t)j+(ddt1)k=3sin3ti+6cos3tj

Therefore, the unit tangent vector is calculated as:

T(t)=r(t)r(t)=

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