Question

Find the unit tangent and unit normal vectors T(t) and N(t). r(t)=

Answer 1

Well looks like you have a space curve. Thus, there is only one direction for your tangent. taking the derivative of r(t), will yield the tangent. so dr/dt (t)=<sqrt(2),e^t,-e^(-t))>. The normal vector will be just the second derivative of r(t). So d^2r/dt (t)=<0,e^t,e^(-t)>. You have to normalize these, which involves you deciding these quantities by their length. I'm sure you can do that. but yet, T(t)=(dr/dt)\||(dr/dt)|| and N(t)=(d^2r/dt^2)/||(d^2r/dt^2||.

Answer 2

Well ||(dr/dt)||=sqrt(2+2e^(2t)) and ||(d^2r/dt^2)||=sqrt(2)e^t.