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Chapter 12.4, Problem 55E
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### Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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BuyFindarrow_forward

### Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem
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# Finding a Binormal Vector In Exercises 47-52, find the vectors T and N and the binormal vector B = T × N for the vector-valued function r(t) at the given value of t. r ( t ) = 4 sin t i + 4 cos t j + 2 t k , t = π 3

To determine

To Calculate: The binomial vector B and vectors T and N for the vector valued function r (t) at the given value of t=0.

Explanation

Given: The vector valued function r(t)= 4sint i + 4cos j+ 2tk, t =π3

Formula Used:

The unit tangent vector, T(t)=r|(t)r|(t)

The principal unit normal vector is N(t)= T|(t)T|(t)

Binormal Vector B=T×N

Calculation:

The derivative of r(t) is r|(t)=4costi4sintj+2k as  ddxsinx=cosx,ddxcosx=sinx

The unit tangent vector, T(t)=r|(t)r|(t)

=4costi4sint j+2k(16cos2t+16sin2t+4)12=4costi4sint j+2k(16+16+4)12=4costi4sint j+2k(20)12=4costi4sint j+2k25=15(2costi2sintj+k). (as sin2t+cos2t=1)Differentiate T(t) with respect to t, we getT|(t)=1/5(2sinti2costj)The principal unit normal vector is N(t)= T|(t)T|(t)or N(t)=(

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