I am working on a project involving structural analysis and I have a small problem regarding vectors. Basically, lets say I have a vector(which represents a beam) going from (0,0,0) to (1,1,1), then that member has its own set of local axis where the local x' axis corresponds to the unit vector of the member itself.

I can work out this out easily enough, but then I need to find a unit vector for the local z' and local y' axis. The only other criteria I have is that the local z' axis will be:

1)parallel to the global xz plane
2)normal to the local x' unit vector

I hope I have made what I am trying to do clear enough. I basically just need these unit vectors so i can relate the local axis (x',y',z') to the global axis (x,y,z) so I can transform forces etc acting in the local system into forces acting in the global system.

In summary, I am trying to find a vector normal to the line that goes from (0,0,0) to (1,1,1) and is also parallel with the xz plane. I figure if I can get this unit vector I can then easily solve for the y' local vector by simply using the cross product of x' and z'.