For a vector $X$ which follows a multinomial Gaussian distribution $N(\vec{0},\Sigma)$, a given vector $b$, and a known scalar value $c$, I would like to calculate the expectation :

$E[X|X^Tb = c]$

That is the expected value of the multivariate variable $X$ given that it will lie on the plane $ X^Tb = c$. I have tried by parametrizing $X$ as $X = \vec{a_0} + t_1 \vec{a_1} ... t_{n-1} \vec{a_{n-1}}$ and calculating the integral $\int_{-\infty}^{\infty} ... \int_{-\infty}^{\infty} xf(x) dt_1 ... dt_{n-1}$, where $f(x)$ is the pdf of the Gaussian, but I end up with an extremely messy formula even when trying to solve in the simple three-dimensional case.

My question is whether there is a known closed form solution for the above expectation and/or if there is a specific parametrization I could use to simplify the solution.