Prove that $X\sim N(\mu,\Sigma)$ is a multivariate random vector if and only if $\langle X,x\rangle$ has a normal distribution for each $x\in\mathbb{R}^n$.
Hint: use Cramér–Wold theorem.
My idea for the $(\Leftarrow)$ part is to use the characteristic function but I do not know how to formalize my reasoning correctly. For the opposite direction I have no ideas on how to use the hint.
Can someone help me?
Edit:
Definition of a standard normal random vector: a random vector $X$ has standard normal distribution in $\mathbb{R^n}$ if it has density: \begin{equation} f(x)=\frac{1}{(2\pi)^{n/2}}e^{-\frac{\|x\|_2^2}{2}} \end{equation} we write $X\sim N(0,I_n)$
Definition: $X$ random vector has general normal distribution in $\mathbb{R}^n$ and we write $X\sim N(\mu,\Sigma)$ for $\mu\in\mathbb{R}^n$ and $\Sigma$ $n\times n$ positive semi-definite matrix if and only if $Z=\Sigma^{-1/2}(X-\mu)\sim N(0,I_n)$