Say I have an $n-$dimensional random vector ${\bf X}$ with iid standard normal entries. Fix a vector $A\in\mathbb{R}^n$.
Question: Does the distribution of ${\bf X}$, conditional on $\langle A,{\bf X}\rangle$ is known? It feels like should be something known. As a further alternative, does, say, $\mathbb{E}[\langle B,{\bf X}\rangle|\langle A,{\bf X}\rangle=t]$ is known? ($B\neq A$)
Any help is appreciated.