Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 1 / 5 / a / Solution

Let $\mathbf{p}=(2,1,1)$, $\mathbf{q}=(1,2,2)$ and $\mathbf{r}=(0,0,3)$. Cross products give plane normals $\mathbf{p}\times \mathbf{r}=(3,-6,0)$ and $\mathbf{q}\times \mathbf{r}=(6,-3,0)$. Their dot product is $36$ and each has norm $3\sqrt{5}$. Thus the acute angle between two planes isThis is also the interior dihedral angle here: perpendicular to the common edge $OR$, the interior wedge lies between the projected directions $(2,1)$ and $(1,2)$. The two outward face normals instead make the supplementary angle.