The problem is to find a unitary matrix A whose first row is a multiple of
a) $(1,1,-i)$ and
b) $\left(\frac{1}{2},\frac{i}{2},\frac{(1-i)}{2}\right)$
Now the first part of a is easy because the rows have to form an Orthonormal set. Thus one gets the usual orthonormalization. However I am stuck as to what to do next, as I am getting just a bunch of equations applying the fact that the dot product has to be zero, especially in part b of the problem.
To complete a unit row $u_1=(u_{11},u_{12},u_{13})$ with $(u_{11},u_{12})\ne 0\ne u_{13}$ to a unitary matrix, we can choose a row $u_2$ proportional to $(\overline{u_{12}},-\overline{u_{11}},0)$ and a row $u_3$ proportional to $\left(u_{11},u_{12},-\frac{|u_{11}|^2+|u_{21}|^2}{\overline{u_{31}}}\right)$. This approach yields matrices $$ \begin{pmatrix} \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{-i}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{6}} & \frac{2i}{\sqrt{6}} \end{pmatrix}\mbox{ and } \begin{pmatrix} \frac{1}{2} & \frac{i}{2} & \frac{1-i}{2} \\ \frac{-i}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 \\ \frac{1}{2} & \frac{i}{2} & \frac{-1+i}{2} \end{pmatrix}, $$ for a) and b), respectively.