let (a,b,c) be perpendicular to (1,2,3)

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)=

t(-2,1,0)+s(-3,0,1)

This is all of the vectors perpendicular to (1,2,3)

they work for all values of s and t.

I hope this helps.

Good luck.