   Chapter 12.4, Problem 38E

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# 57095-12.4-38E-Question-Digital.docxFinding Vectors In Exercises 37–42, find T(t), N(t), a T , and a N at the given time t for the space curve r(t). [Hint: Find a(t), T(t), a T , and a N . Solve for N in the equation a(t) = a N T + a N N.]Vector-valued function    Time r ( t ) = cos   t i + sin   t j + 2 t k t = π 3

To determine

To calculate: The vectors T(t),N(t),aT and aN, of the curve, r(t)=costi+sintj+2tk at t=π3.

Explanation

Given:

r(t)=costi+sintj+2tk and t=π3.

Formula used:

If r(t) is a curve then,

T(t)=r(t)r(t),N(t)=T(t)T(t),aT=ddt[v(t)] and aN=v(t)T(t).

And a(t)=aTT+aNN

Calculation:

Consider the curve, r(t)=costi+sintj+2tk.

Now, r(t)=sinti+costj+2k

And,

r(t)=(sint)2+(cost)2+(2)2=1+4=5

Now, find T(t)=r(t)r(t).

So,

T(t)=sinti+costj+2k5=5sint5i+5cost5j+255k

Now, at t=π3

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