What is the unit tangent vector to the curve at the specified value of the parameter?

Question

r(t)=#t^3hati+8t^2hatj#, t=2

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Answer 1 · 2017-10-15T21:05:02

Given: #vecr(t) = t^3hati + 8t^2hatj#

The tangent vector is:

#vec(T(t)) = 3t^2hati+16thatj#

Evaluate at #t = 2#:

#vec(T(2)) = 12hati+32hatj#

We can obtain the unit vector by dividing my the magnitude:

#|vec(T(2))| = sqrt((12)^2 + (32)^2)= 4sqrt73#

#hatT(2) = 4sqrt73/73hati+8sqrt73/73hatj#