
Find a number s that garantees that the vectors
i, i+j+k, and i+2j+sk will all be parallel to the
same plane.

Three vectors are on a plane when and only when they generate
no volume, i.e. their triple scalar product is 0.
Let u,v and w be the three given vectors, 
> u := [1,0,0]: v := [1,1,1]: w := [1,2,s]:
their triple scalar product is, 
> tsp := dotprod(u,crossprod(v,w));
_ tsp := 2 + s
s is real (not complex) so its congugate is itself. Hence the value of s that makes tsp = 0 is 
> s = 2: