I have three (N x 1) complex vectors, a, b and c. I know the following conditions: (1) a and b are orthonormal (but length of c is unknown) (2) c lies in the same 2D plane as a and b (3) aHc = x (purely real, known) (4) bHc = iy (purely imaginary, unknown) where (.)H denotes Hermitian (conjugate) transpose, i is the imaginary unit and x,y are real numbers. Given that I know x, can I deduce y? My hunch is that (without the "purely real/imaginary" statements), these conditions would define y up to an arbitrary complex phase, but the "purely real/imaginary" conditions allow the phase to be known too. However, my reasoning relies on there being some sense of "angle" between a and c and between b and c... such that these angles sum to 90° for the orthonormality condition (1). I don't know if this is valid.