Answer
$\sqrt 2 i+e^tj-e^{-t}k$, $e^t j+e^{-t}k$ , $e^t+e^{-t}$
Work Step by Step
As we are given $r(t)=\sqrt 2 ti+e^tj+e^{-t}k$
Need to determine velocity vector, acceleration vector and speed.
We have
$v(t)=r'(t)$ and $a(t)=v'(t)$ and speed is the magnitude of the velocity vector, that is $s(t)=|v(t)|$.
$v(t)=r'(t)=\sqrt 2 i+e^tj-e^{-t}k$ [ given: $r(t)=\sqrt 2 ti+e^tj+e^{-t}k$]
Also, $a(t)=v'(t)=e^t j+e^{-t}k$
Now, $s(t)=|v(t)|=\sqrt {(\sqrt 2)^2+(e^t)^2+(-e^{-t})^2}=e^t+e^{-t}$
