Past exam of the condensed matter physics course of the University of Oxford / 2017 / 1 / c / ii / Solution

The $(100)$ normal vector is $(1,0,0)$. Its normalized dot product with $(1,1,1)$ is $1/\sqrt{3}$, soThis is the acute angle between the lattice planes, rather than the complementary angle between a plane and the other plane's normal.