   Chapter 12, Problem 47RE

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# Finding the Principal Unit Normal Vector In Exercises 47-50, find the principal unit normal vector to the curse at the specified value of the parameter. r ( t ) = 2 t i + 3 t 2 j , t = 1

To determine

To calculate: The principal unit normal vector to the curve r(t)=2ti+3t2j at the specified value of the parameter where, t=1.

Explanation

Given:

The vector-valued curve is: r(t)=2ti+3t2j and the parameter is t=1.

Formula used:

Principal unit normal vector:

N(t)=T(t)T(t)

Unit Tangent Vector:

T(t)= r(t)r(t)

The magnitude of a vector x=ai+bj is x=a2+b2.

Calculation:

Consider the provided vector-valued curve,

r(t)=2ti+3t2j

Differentiate the vector r(t).

r(t)=2i+6tjr(t)=4+36t2

At t=1

r(1)=4+36(1)2=4+36=210

Unit Tangent Vector:

T(t)= r(t)r(t)

T(t)=24+36t2i+6t4+36t2j

At t=1

T(1)= r(1)r(1) Now, calculate the unit tangent vector.

T(1)= r(1)r(1)= 2210i+6210j= 110i+310j

Now,

T(t)=22(4+36t2)32(36×2t)i+64+36t26t(36×2t24+36t2)(4+36t2)j=72t(4+36t2)32i+6(

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