In a cubic crystal, which value is closest to the angle between the [100] and [111] directions?

Answer: C. About 54.7°

In cubic crystals, the angle between two directions can be found from the dot product. For [100] and [111], cosθ = (1×1 + 0×1 + 0×1)/(1 × √3) = 1/√3 ≈ 0.577, so θ ≈ 54.7°. About 35.3° is the angle between [110] and [111] (cosθ = 2/√6), and about 45.0° is the angle between [100] and [110]. Because the [hkl] direction is normal to the (hkl) plane in cubic crystals, 54.7° is also the angle between the (100) and (111) planes, familiar as the slope of the {111} sidewalls that appear when (100) silicon is etched in an alkaline solution.