Geometric Explanation of R-Squared Suppose we have two orthogonal features $x_1$ and $x_2$. If we run univariate regression of $y$ on $x_1$ or $x_2$ and get $R^2, $$r_1$ and $r_2$ and we run multivariate regression of $y$ on both $x_1$ and $x_2$ and get the $R^2$ which is denoted by $r$. A well-known result is that $r = r_1 + r_2$. I am interested in how can we connect this result to the cosine angel. We know that the $cos\theta$ is the correlated between two vectors that have an angel $\theta$. An example is that if we represent $x_1 = (1, 0, 0)$ and $x_2 = (1, 0, 0)$. We further write $y = (a, b, 1)$. We know the regression $R^2$ is: $r_1 = \frac{a}{\sqrt{(a^2 + b^2 + 1)}}$ and $\frac{b}{\sqrt{(a^2 + b^2 + 1)}}$.
If we project $y$ on the space spanned by $x_1$ and $x_2$, we know the projection can be written as $z = (a, b, 0)$. The correlation between this projection and $y$ is $\frac{\sqrt{(a^2+b^2)}}{\sqrt{(a^2 + b^2 + 1)}}$ which is $r$. However it does not seem that this is the sum of $r_1$ and $r_2$. Did I miss anything?

You can find a detailed examination of the geometric properties of multiple linear regression in O'Neill (2019).  As you can see from that paper, it is indeed possible to determine the coefficient of determination in the regression using the cosines of the angles between all the relevant vectors.  You can also see how the coefficient of determination in a multiple linear regression relates to the corresponding results in individual simple linear regressions using the same explanatory vectors.
Rather than running through your example, I will point you to the section on combining results in the model with two explanatory variables (pp. 10-11).  This paper should assist you to understand the relevant geometric properties and the angular formulae for relevant regression quantities.
