Chapter 4.4, Problem 37E

[M] Exercises 37 and 38 concern the crystal lattice for titanium, which has the hexagonal structure shown on the left in the accompanying figure. The vectors $\left[\begin{array}{c}2.6\\ -1.5\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 3\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 4.8\end{array}\right]$ in ℝ³ form a basis for the unit cell shown on the right. The numbers here are Ångstrom units (1 Å = 10⁻⁸ cm). In alloys of titanium, some additional atoms may be in the unit cell at the octahedral and tetrahedral sites (so named because of the geometric objects formed by atoms at these locations).

The hexagonal close-packed lattice and its unitcell.

37. One of the octahedral sites is $\left[\begin{array}{c}1/2\\ 1/4\\ 1/6\end{array}\right]$, relative to the lattice basis. Determine the coordinates of this site relative to the standard basis of ℝ³.