   Chapter 12.4, Problem 54E

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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 ) = 2 e t i + e t cos t j + e t sin t k , t = 0

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)=2eti+etcostj+etsintk, t=0.

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 differentiation of r(t) is  r|(t)=2eti+(etcostetsint)j+(etsint+etcost)k

Hence, the unit tangent vector, T(t)=r|(t)r|(t)

or T(t)=2eti+(etcostetsint)j + (etsint+etcost)ket4+(costsint)2+(sint+cost)2=2i+(costsint)j+(sint+cost)k4+1+1 (as sin2t+cos2t=1.)=16[2i+(costsint)j+(sint+cost)k]Differentiate T(t) with respect to t, we getT"(t)=16[(sintcost)j+(costsint)k]The principal unit normal vector, N(t)=T|(t)T|(t)or N(t)=(sin</

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