Answer
$\sqrt 2 i+e^tj-e^{-t}k$, $e^t j+e^{-t}k$ , $e^t+e^{-t}$
Work Step by Step
Given: $r(t)=\sqrt 2 ti+e^tj+e^{-t}k$
Our aim is to calculate the velocity vector, acceleration vector and speed.
In order to calculate the all above terms we will use formulas, such as:
$v(t)=r'(t)$ and $a(t)=v'(t)$ and speed is the magnitude of the velocity vector, that is $s(t)=|v(t)|$.
Now,
$v(t)=r'(t)=\sqrt 2 i+e^tj-e^{-t}k$
$a(t)=v'(t)=e^t j+e^{-t}k$
$s(t)=|v(t)|=\sqrt {(\sqrt 2)^2+(e^t)^2+(-e^{-t})^2}=e^t+e^{-t}$
Hence, the required answers are:
$\sqrt 2 i+e^tj-e^{-t}k$, $e^t j+e^{-t}k$ , $e^t+e^{-t}$
