Since the phasors are rotating at the same speed, and their amplitudes and phases are constant, we can solve it by considering the phasors as being static:
letting phasor 1 point along the y-axis, it's endpoint is given by the position vector where is the x ordinate of the origin of the first phasor, and i,j, and k are the basis vectors.
then the endpoint of the second phasor is given by . where is the distance between the origins of the two phasors, and is the phase diffference between them.
Then the vector form the enpoint of phasor 1 to the endpoint of phasor 2 (your blue line) is given by:
From this you can find the vector equation of the line by using the parameter :
Where r is also a vector.
Then you can find the equation of the plane passing through x_0 that is parrallel to the y-z plane, and find the intersection of the line with this plane.
This will give you the endpoint of your third phasor, from which you can easily work out the amplitude and phase.