So suppose $U\_1,U\_2,U\_1 \neq cU\_2$ are unitary operators in finite dimesional complex vector space and $\nu\_1,\nu\_2$ are two complex numbers. What should I require from $\nu\_1,\nu\_2$ to make $\nu\_1U\_1 + \nu\_2U\_2$ also unitary?

My attempt:

Let $U = \nu\_1U\_1 + \nu\_2U\_2$. 
$$
UU^{\dagger} = ( |\nu\_1|^2+|\nu\_2|^2)I + 2Re(\nu\_1\bar{\nu\_2}U\_1U\_2^\dagger)
$$
I want it to be equal to $I$, so:
$$
Re(\nu\_1\bar{\nu\_2}U\_1U\_2^\dagger) = \frac{1- |\nu\_1|^2-|\nu\_2|^2}{2}I
$$
And here I stucked. Help please.