   Chapter 12.4, Problem 40E

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# 57095-12.4-40E-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 ) = ( 2 t − 1 ) i + t 2 j − 4 t k t = 2

To determine

To calculate: The vectors T(t),N(t),aT and aN, of the curve, r(t)=(2t1)i+t2j4tk at t=2.

Explanation

Given:

r(t)=(2t1)i+t2j4tk and t=2.

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)=(2t1)i+t2j4tk.

Now, r(t)=2i+2tj4k

And,

r(t)=(2)2+(2t)2+(4)2=4t2+20=2t2+5

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

So,

T(t)=2i+2tj4k2t2+5=1t2+5i+tt2+5j2t2+5k

Now, at t=2.

T(2)=1(2)2+5i+2(2)2+5j2(2)2+5k=13i+23j23k

For N(t)

T(t)=2t2(t2+5)32i+t2+5(1)(t)2t2t2+5(t2+5)2j+4t2(t2+5)32k

At t=2

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