A linear combination of iid normal vectors by row-vector weights is scalar (univariate) normal

Suppose $Y_1,Y_2$ and $Y_3$ are independent and identically distributed $p$-variate random vectors having normal distribution. Then the distribution of $a_1Y_1+a_2Y_2+a_3Y_3$, where $a_1,a_2,a_3$ are vectors of order $1\times p$, is: