Consider the vector function given below.

r (t) = (3t^2, sin (t) - tcos (t), cos (t) + tsin (t)), t>0

Do the following

(a) Find the unit tangent and unit normal vectors T (t) and N (t)

T (t) =

N (t) =

(b) Find the curvature

k (t) =

The tangent vector is by definition the derivative of r (t) with respect to t:

T' = dr/dt =

The unit vector T = T'/|T'| = / sqrt (36t^2 + t^tsin (t) ^2 + t^2cos (t^2))

T = / (t*sqrt (37)) = / sqrt (37)

Now the normal unit vector N is perpendicular to r/|r| and T. It is the second derivative of r/|r| with repsect to time

N' = d^2r/dt^2 =

N = N'/|N'| = / sqrt (36 + sin^2t + 2tsin (t) cos (t) + t^2cos^2t + cos^2 (t) - 2tcos (t) sin (t) + t^2sin^2t)

N = / sqrt (37 + t^2)