I think v=(-32/19,54/19,31/19) w=(-6/19,3/19,-12/19) is be the solution.

you have to solve 4 equations simultaniously:

w=a*(2,-1,4) for some a

w=(2a,-a,4a)

also let v=(x,y,z)

v dot u=0 => 2x-y+4z=0

if you now use that v+w=(-2,3,1) you have 4 equations and 4 unknowns.