Since they are perpendicular their dot product must be zero
now we need to solve this underdetermined system of equations.
let c=s b=t then a=-2t-3s
so we get the vector (-2t-3s,t,s)=(-2t,t,0)+(-3s,0,s)=
This is all of the vectors perpendicular to (1,2,3)
they work for all values of s and t.
I hope this helps.