  In this letter we derive the $(n-1)$-dimensional distribution corresponding
to a $n$-dimensional i.i.d. Normal standard vector $Z=(Z_1,Z_2,\ldots,Z_n)$
subjected to the weighted sum constraint $\sum_{i=1}^n w_i Z_i=c$, $w_i\neq 0$.
We first address the $n=2$ case before proceeding with the general $n\geq 2$
case. The resulting distribution is a Normal distribution whose mean vector
$\mu$ and covariance matrix $\Sigma$ are explicitly derived as a function of
$w_1,\ldots,w_n,c$. The derivation of the density relies on a very specific
positive definite matrix for which the determinant and inverse can be computed
analytically.
