correlation range between y and two variables x1 and x2 What is the correlation coefficient (or range) $corr(y,\hat{y})$ for the regression $\hat{y}=ax_1+bx_2+c$ given that the correlation between $x_1$ and $y$ is 0.5 and the correlation between $x_2$ and $y$ is 0.5,and the correlation between $x_1$ and $x_2$ is 0. And what about the $R^2$?

Given a vector of $n$ ground truth values $\mathbf{y} \in \mathbb{R}^n$ and a vector of predicted values $\mathbf{\hat{y}}\in \mathbb{R}^n$, you can write the covariance between the predictions and the ground truth as the dot product $\mathbf{\hat{y}}^\intercal \mathbf{y}$ (we suppose that the data is centered around zero to simplify the notations).
Then if you replace $\mathbf{\hat{y}}$ by $a\mathbf{x_1} +b\mathbf{x_2}$ you may notice that the covariance of the prediction with your ground truth is simply the sum of the individual covariances, which is also true for the correlation :
$\mathbf{\hat{y}}^\intercal \mathbf{y} = (a\mathbf{x_1} +b\mathbf{x_2} )^\intercal y = (a\mathbf{x_1})^\intercal y + (b\mathbf{x_2})^\intercal y.$
To be specific, in your case, the correlation is 1.
