I want to find
$$ E[\Phi_1 | c_1 \Phi_1 + c_2 \Phi_2 + c_3\Phi_3 = x] $$
as well as (if possible?)
$$ E[\alpha_1 \Phi_1 + \alpha_2\Phi_2 | c_1 \Phi_1 + c_2 \Phi_2 + c_3\Phi_3 = x] $$
where $c_1, c_2, c_3, \alpha_1, \alpha_2$ are constants, given that:
$$ \begin{bmatrix} \Phi_1 \\ \Phi_2 \\ \Phi_3 \end{bmatrix} \sim \mathcal{N}\left( \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{12} & \sigma_{22} & \sigma_{23} \\ \sigma_{13} & \sigma_{23} & \sigma_{33} \\ \end{bmatrix} \right) $$
Let $X = c_1 \Phi_1 + c_2 \Phi_2 + c_3\Phi_3$. I have found that $E[X] = 0$ and $\text{var}[X] = c_1^2\sigma_1^2 + c_2^2 \sigma_2^2 + c_3^2 \sigma_3^2 + 2c_1c_2\sigma_{12} + 2c_1c_3\sigma_{13} + 2c_2c_3\sigma_{23}$, although I'm not sure these help me in any way.
I'm doing this for a research project I'm working on. So much of the information out there seems to be on independent R.V.s. Is there a typical pattern to follow in order to go about solving general problems like these? Are there any specific references that I should look up?