Answer
$\lt 2t, t \sin t, t\cos t \gt$, $\lt 2, t \cos t+\sin t, \cos t-t \sin t \gt$
, $t\sqrt 5$
Work Step by Step
As we are given that $r(t)=\lt t^2, \sin t- t \cos t, \cos t+ \sin t \gt$
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 $s(t)=|v(t)|$.
$v(t)=r'(t)=\lt 2t, \cos t+ t \sin t-\cos t, -\sin t+ t\cos t +\sin t\gt=\lt 2t, t \sin t, t\cos t \gt$ [given: $r(t)=\lt t^2, \sin t- t \cos t, \cos t+ \sin t \gt$]
Also, $a(t)=v'(t)=\lt 2, t \cos t+\sin t, \cos t-t \sin t \gt$
Thus, $s(t)=|v(t)|=\sqrt {(2)^2+(t \cos t+\sin t)^2+( \cos t-t \sin t)^2}=\sqrt {4t^2+(t\sin t)^2+( t \cos t)^2}$
$=t\sqrt 5$
