Suppose $\vec{r}(t) = \cos\!\left(\pi t\right)\boldsymbol{i}+\sin\!\left(\pi t\right)\boldsymbol{j}+2t\boldsymbol{k}$ represents the position of a particle on a helix, where $z$ is the height of the particle.

(a) What is $t$ when the particle has height $8$?
$t =$

(b) What is the velocity of the particle when its height is $8$?
$\vec{v} =$

(c) When the particle has height $8$, it leaves the helix and moves along the tangent line at the constant velocity found in part (b). Find a vector parametric equation for the position of the particle (in terms of the original parameter $t$) as it moves along this tangent line.
$L(t) =$