Proving two random vectors are following the same distribution using characteristic function?

Question

Suppose we have two centered Gaussian random vectors $X$ and $Y$ I.I.D.

Find the distribution of $X_a=X\cos(a)+Y\sin(a)$ and $Y_a=-X\sin(a) + Y\cos(a)$ with $a$ being in $[0,2\pi)$.

What I tried to do is write the characteristic function of $X_a$ and prove that it's something obvious (like maybe the same law as $X$).

\begin{align} \phi_{X_a}(u)&=E[\exp(i\langle u,X\cos(a)+Y\sin(a)\rangle)]\\ &=E[\exp(i\langle u,X\cos(a)\rangle)]\cdot E[\exp(i\langle u,Y\sin(a)\rangle)] \end{align}

but I got stuck right there and didn't know how to proceed. Any hints?

Answer 1

If $a$ is a constant, then $X_a = X \cos(a) + Y \sin(a)$ is a linear combination of two independent zero-mean Gaussian R.V.s, which is a zero-mean Gaussian R.V. with the variance $\sigma_{X_a}^2 = \cos^2(a) \sigma_X^2 + \sin^2(a) \sigma_Y^2$. The same goes for $Y_a$. If $X,Y\sim\mathcal{N}(0,1)$, then $X_a,Y_a\sim\mathcal{N}(0,1)$ too.