Find the unit tangent vector for the curve r(t)=⟨4t,cos(9t),sin(9t)⟩. Write the exact answer. Do not round.

Question

Im unsure how to go about this

Answer 1

To find the unit tangent vector, you take the derivative of each component with respect to t, then divide by the length of the vector. We have:

r'(t) = <4, -9sin(9t), 9cos(9t)>

The length of r'(t) is:

||r'(t)|| = sqrt(16 + 81sin²(9t) + 81cos²(9t)) = sqrt(16 + 81) = sqrt(97)

Therefore, the unit tangent vector is:

r'(t)/||r'(t)|| = <4/sqrt(97), -9sin(9t)/sqrt(97), 9cos(9t)/sqrt(97)>