Find the velocity and acceleration vector and the equation of the tangent line for the following curve, at t=0:
\(\vec{c}(t)=(t\sin t,t\cos t,\sqrt{3}t)\)
Velocity vector:\(\vec{v}(t)=D\vec{c}(t)=(t\cos t+\sin t)\vec{i}+(\cos t-t\sin t)\vec{j}+\sqrt{3}\vec{k}\)
\(\vec{v}(0)=\vec{j}+\sqrt{3}\vec{k}\)
Acceleration vector:\(\vec{a}(t)=D\vec{v}(t)=(2\cos t-t\sin t)\vec{i}-(2\sin t+t\cos t)\vec{j}\)
\(\vec{a}(0)=2\vec{i}\)
Equation of tangent line:\(\vec{l}(t)=t\vec{j}+\sqrt{3}t\vec{k}\)
