Solution
= Solution
The $(100)$ <normal vector> is $(1,0,0)$. Its normalized dot product with $(1,1,1)$ is $1/\sqrt3$, so
$$
\boxed{\phi=\cos^{-1}\!\left(\frac1{\sqrt3}\right)=54.7356^\circ.}
$$
This is the acute angle between the <lattice planes>, rather than the complementary angle between a plane and the other plane's normal.