Answer
$2ti+2j+\dfrac{1}{t}k$, $2i-\dfrac{1}{t^2}k$ , $2t+\dfrac{1}{t}$
Work Step by Step
As we are given that $r(t)=t^2i+2t j+\ln t k$
Need to determine the 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 $s(t)=|v(t)|$.
$v(t)=r'(t)=2ti+2j+\frac{1}{t}k$ [ given : $r(t)=t^2i+2t j+\ln t k$]
Now, $a(t)=v'(t)=2i-\frac{1}{t^2}k$
Thus, $s(t)=|v(t)|=\sqrt {(2t)^2+(2)^2+(\frac{1}{t})^2}=2t+\frac{1}{t}$
