   Chapter 12.4, Problem 56E

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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 ) = 3 cos 2 t i + 2 sin 2 t j + t k , t = π 4

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)= 3cos2t i + 3sin2t j+ tk, t =π4

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)= 6sin2t i+ 6cos2t j + k

The unit tangent vector, T(t)=r|(t)r|(t)=6sin2t i+ 6cos2tj +k36+1=137(6sin2t i+ 6cos2t j +k). (as sin2t+cos2t=1)Differentiate T(t) with respect to t, we getT|(t)=137(12cos2t i 12sin2t j)The principal unit normal vector is N(t)= T|(t)T|(t)or N(t)=12(cos2ti+sin2t j)144 , (as sin2t+cos2t=1)=(cos2t

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