conditional expectation of normal random variables

Question

I am not the best when it comes to conditional expectation and I am currently going over an economic/finance theory paper and they have the following statement:

$V_D = \delta_D + d_D$ x $\delta_F + \eta$

$V_F = d_F$ x $\delta_D + \delta_F + \nu$

Assume that $\nu=0$, $d_F=1$, and $d_D\in(0,1)$. $\delta_D$ and $\delta_F$ are normally distributed ~ $N(0,1)$ and $\eta$ is also normally distributed ~ $N(0,\sigma^2_{_\eta})$

Let $\omega_F=\delta_F+B_Fu_F$

I would like to know to prove the following:

The authors show that using normal random variable properties $E[V_D|\delta_D,\omega_F] = \delta_D + \frac{d}{(1+B^2_F\sigma^2_{uF}}\omega_f $

also what should be the value of $VAR[\nu_D|\delta_D,\omega_F]$?

I hope this is clear...if anything is missing please let me know.