Suppose I have a $d$-dimensional multivariate Gaussian random variable $X \sim \mathcal{N}(\mathbf{t}, I_d)$ ($I_d$ is the $d$-dimensional identity matrix) and $m$ vectors $\mathbf{w}_1, \mathbf{w}_2, ..., \mathbf{w}_m \in \mathbb{R}^d$ all normalized so that $\Vert \mathbf{w}_i \Vert_2^2 = 1$ for all $i \in [m]$. Then $$ Y_i = \mathbf{w}_i^TX \sim \mathcal{N}(\mathbf{w}_i^T\mathbf{t}, 1) $$ can be easily shown based on properties of multivariate Gaussian random variables.
However, what will the joint distribution of $Y = [Y_1, Y_2, ..., Y_m]^T$ be?
So far I have only managed to show that with $m = d= 2$ and $\mathbf{t} = \mathbf{0}$, $$ Y = [Y_1, Y_2]^T \sim \mathcal{N}( \mathbf{0}, \begin{bmatrix} 1&\mathbf{w_1}^T\mathbf{w}_2\\\mathbf{w_1}^T\mathbf{w}_2&1 \end{bmatrix} ) $$ based on the results from O'Donell's Analysis of Boolean Functions (p.328, https://www.cs.tau.ac.il/~amnon/Classes/2016-PRG/Analysis-Of-Boolean-Functions.pdf).
And I guess with $X$ having mean $\mathbf{t} \neq \mathbf{0}$, the resulting $Y$'s covariance matrix should remain the same as what I had in the 2-dimensional case, only with its mean changing to $[\mathbf{w}_1^T\mathbf{t}, ..., \mathbf{w}_m^T\mathbf{t}]^T$, but I can't really extend my results to more general cases.
Does anyone have a better idea on this?