I am trying to work through an online solution to Exercise 4.9 in Nielsen and Chuang's "Quantum Computation and Quantum Information:
The question is: Explain why any single qubit unitary operator may be written in the form $$ U= \begin{bmatrix} e^{i(\alpha - \frac{\beta}{2} - \frac{\delta}{2})} cos(\frac{\gamma}{2}) & -e^{i(\alpha - \frac{\beta}{2} + \frac{\delta}{2})} sin(\frac{\gamma}{2})\\ e^{i(\alpha + \frac{\beta}{2} - \frac{\delta}{2})} sin(\frac{\gamma}{2}) & e^{i(\alpha + \frac{\beta}{2} + \frac{\delta}{2})} cos(\frac{\gamma}{2}) \end{bmatrix} $$
It seems from the solutions online that I've looked at that we should prove that the rows/columns of a unitary matrix are orthonormal. We can do this by
$$ U= \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$
and then $U^{\dag}U=I$ tells us that
$$ \begin{bmatrix} aa^* + bb^* & ac^* + bd^* \\ ca^* + db^* & cc^* + dd^* \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $$
And this shows that the columns/rows are orthonormal. I know that if we let $$v_{1} = \begin{bmatrix} a & b \end{bmatrix} $$
and we let $$v_{2}= \begin{bmatrix} c & d \end{bmatrix}$$
then these are orthonormal, but not sure how it proves that the columns are orthonormal?
In addition, not sure how to finish the question and prove that every single qubit unitary operator can be written in the form stated in the question?