(a) Find a parametric equation for the line through the point $\left(2,4,2\right)$ and in the direction of $a\vec i +b\vec j+c\vec k$.
$\vec r(t) =$

(b) Find conditions on $a,b,c$ so that the line you found in part (a) goes through the origin. (Be sure you can give a reason for your answer.) Then use your work to give two distinct triples $a,b,c$ that result in the line passing through the origin.
Choice 1: $a =$ , $b =$ , $c =$ ;
Choice 2: $a =$ , $b =$ , $c =$ .