   Chapter 12.4, Problem 16E

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# 57095-12.4-16E-Question-Digital.docxFinding the Principal Unit Normal Vector In Exercises 13-20, find the principal unit normal vector to the curve at the specified value of the parameter. r ( t ) = π   cos   t i + π   sin   t j , t = π 6

To determine

To-calculate: The principal unit normal vector to the curve r(t)=πcosti+πsintj at t=π6.

Explanation

Given:

The vector, r(t)=πcosti+πsintj and a point t=π6.

Formula used:

Unit normal vector N(t) to the curve, r(t) is given by,

N(t)=T(t)T(t) where T(t)=r(t)r(t) is the unit tangent vector.

Calculation:

Consider the vector, r(t)=πcosti+πsintj.

Now, use the formula, T(t)=r(t)r(t) to find the tangent vector to the curve,r(t)=πcosti+πsintj.

First calculate, r(t).

So, r(t)=πsinti+πcostj.

Now calculate, r(t).

So,

r(t)=(πsint)2+(πcost)2=π2(sin2t+cos2t)=π

Now put, r(t)=πsinti+πcostj and r(t) in the formula, T(t)=r(t)r(t)

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