Find the angle between the main diagonal of a cube and a skew diagonal of a face of the cube I was told it was $90$ degrees, but then others say it is about $35.26$ degrees. Now I am unsure which one it is.


If we assume the cube has unit side length and lies in the first octant with faces parallel to the coordinate planes and one vertex at the origin, then the the vector $(1,1,0)$ describes a diagonal of a face, and the vector $(1,1,1)$ describes the skew diagonal.
The angle between two vectors $u$ and $v$ is given by: 
$$\cos(\theta)=\frac{u\cdot v}{|u||v|}$$
In our case, we have 
$$\cos(\theta)=\frac{2}{\sqrt{6}}\quad\Longrightarrow\quad\theta\approx 35.26$$
