Answer
$\theta =\dfrac{\pi}{2}$
Work Step by Step
We know that the velocity and acceleration is defined as: velocity is given as: $v(t)=r'(t)$ and acceleration is given as: $a(t)=v'(t)$
In the given problem, we have
$r=\lt 3t+1, \sqrt 3t ,t^2 \gt$
Now, $v(t)=r'(t)=\lt 3, \sqrt 3 ,2t \gt \\ \implies v(0)=\lt 3, \sqrt 3 ,0\gt = 3i +\sqrt 3 j$
and $a(t)=v'(t)=\lt 0,0,2 \gt \\ \implies a(0)=\lt 0,0,2 \gt= 2k$
We have found that the velocity $v(0)$ lies in xy plane and acceleration $a(0)$ lies in the direction of z-axis. This implies that $\theta =\dfrac{\pi}{2}$
