Normally distributed vectors and inner product

Question

Suppose that we have two random vectors $X_{1\times n}$ and $Y_{1\times n}$, where $X\sim N(\mu_x,\sigma_{x}^{2})$ and $Y\sim N(\mu_y,\sigma_{y}^{2})$, with N standing for the standard notation of the normal distribution $\mu_x,\mu_y$ are constants and $\sigma_{x}^{2}, \sigma_{y}^{2}>0$. Can we show that the inner product of the vectors $<X,Y>=XY^{'}$ also follows a normal distribution (where the exponential ${'}$ denotes the transpose of the random vector $Y$)? Does the same hold if $X$ and $Y$ is random variables?

Answer 1

Note: The question has seen a metamorphosis after I posted that answer.

Take $n=1$. Then the inner product is simply the product $XY$. So you are asking if product of two normal random variables is normal. This is false. In particular this is false when $X=Y$ because $X^{2} \geq 0$.