the angle between directions
Two crystallographic directions meeting at a point subtend an angle, and you frequently need its value — for instance to check the angle between a slip direction and an applied stress, or between two rows of atoms seen in a micrograph. The angle between directions is found the same way you would find the angle between any two vectors: with a dot product.
In a cubic crystal the formula is as tidy as it gets. For directions [u1 v1 w1] and [u2 v2 w2], cos theta = (u1 u2 + v1 v2 + w1 w2) / (sqrt(u1^2 + v1^2 + w1^2) times sqrt(u2^2 + v2^2 + w2^2)). Try [100] and [111]: the top is 1, the bottom is 1 times sqrt(3), so cos theta = 0.577 and theta = 54.7 degrees. Try [110] and [111]: the top is 2, the bottom is sqrt(2) times sqrt(3) = sqrt(6), so cos theta = 2/sqrt(6) = 0.816 and theta = 35.3 degrees.
Notice that these are exactly the same numbers that come out of the interplanar-angle formula. That is no accident: in a cubic crystal the direction [hkl] is perpendicular to the plane (hkl), so the angle between two directions equals the angle between the two like-indexed planes, and one formula does both jobs. But this neat overlap is again a cubic-only gift — in non-cubic systems directions and planes must be handled with the proper metric and no longer share the simple formula.
In a cubic crystal the angle between [100] and [110] is arccos(1/sqrt(2)) = 45 degrees, and between [111] and [100] it is arccos(1/sqrt(3)) = 54.7 degrees — identical to the corresponding plane angles, because in cubic each direction is the normal of its like-indexed plane.
A vector dot-product gives the angle between two directions — coinciding with the plane formula only in cubic.
Like the plane-angle formula, this dot-product holds only for cubic crystals. Directions and planes share the same numbers there purely because [hkl] is the normal of (hkl) in cubic — not in general.