   Chapter 12, Problem 52RE

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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 ) = 3 cos 2 t i + 3 sin 2 t j + k , t = π 6

To determine

To calculate: The tangential and normal component of acceleration for the space curve r(t) where r(t)=3cos2ti+3sin2tj at t=π6.

Explanation

Given:

The vector-valued function is: r(t)=3cos2ti+3sin2tj at t=π6.

Formula used:

The tangential and Normal acceleration are given as:

aT=vavaN=v×av

Calculation:

Consider the vector-valued function,

r(t)=3cos2ti+3sin2tj

Differentiate the function with respect to t to find the velocity vector.

v(t)=r(t)=6sin2ti+6cos2tj

And, v(t)=6

Differentiate the velocity vector with respect to t,

a(t)=v'(t)=12cos2ti12sin2tj

Thus,

v

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