I define the vector function $\vec{r}(t)$ by

r[t_] = {t, Cos[t], Sin[t]}

The command r'[t] gives the vector tangent to the curve drawn by $r(t)$ for any specified value of $t$.

I can generate the curve for $r(t)$ on the interval $[0, 2\pi]$ via

ParametricPlot3D[{t, Cos[t], Sin[t]}, {t, 0, 2 Pi}]

What I would like to do is draw a tangent vector whose tail actually touches the curve. For example, I would like to draw the vector described by $r^\prime(0)$ with its tail at $r(0)$. Even better would be to wrap this inside Manipulate and draw the tangent vector for any $t \in [0, 2\pi]$. How can I accomplish either of these goals?