Independence of Gaussian projections Suppose  $x=(x_1,x_2)$, whose entries are sampled iid from $N(0,1)$. Let $a=(a_1,a_2)\in\mathbb{R}^2$ and $b=(b_1,b_2)\in\mathbb{R}^2$ be two unit vectors such that $a^{\top}b=a_1b_1+a_2b_2=0$.  Let $X=a^{\top}x$ and $Y=b^{\top}x$. Then $X\sim N(0,1)$ and $Y\sim N(0,1)$. How do we show that $X$ and $Y$ independent?

Since this is likely a homework problem, I won't provide an answer but give suggestions as to how to construct the desired proof.
A standard definition of joint normality of two random variables $W$ and $Z$ is that $W$ and $Z$ are said to be jointly normal if their linear combination $\alpha W + \beta Z$ is a normal random variable for all choices of $\alpha, \beta \in \mathbb R$. Choosing $(\alpha,\beta)$ as $(1,0)$ or $(0,1)$ shows us that when $W$ and $Z$ are jointly normal, they are marginally normal as well.
You have constructed $X$ and $Y$ as linear combinations of $X_1$ and $X_2$ and deduced that $X$ and $Y$ are normal random variables, indeed $N(0,1)$ random variables. Can you use the above definition of jointly normal random variables to show that $X$ and $Y$ are jointly normal as well? That is, can you write down a proof that every linear combination of $X$ and $Y$ is a normal random variable? Don't shirk this chore: normal random variables are not necessarily jointly normal. Once you have proven joint normality of $X$ and $Y$, find $\operatorname{cov}(X,Y)$, write down the joint pdf of $X$ and $Y$ and see if it factors into the product of the marginal pdfs of $X$ and $Y$. If so, $X$ and $Y$ are independent.
