determine the crystallographic directions of vectors b, c, and d shown in the following cubic unit cell…

Explanation:

Step1: Find vector - end points for B

Assume origin ((0,0,0)) and end - point of B is ((\frac{2}{3},0,0)). Subtract origin coordinates from end - point coordinates: (\frac{2}{3}-0=\frac{2}{3}), (0 - 0=0), (0 - 0=0). Multiply by 3 to get integer indices ([2\ 0\ 0]).

Step2: Find vector - end points for C

Assume origin ((0,0,0)) and end - point of C is ((0,1,\frac{1}{3})). Subtract origin coordinates: (0 - 0=0), (1 - 0 = 1), (\frac{1}{3}-0=\frac{1}{3}). Multiply by 3 to get integer indices ([0\ 3\ 1]).

Step3: Find vector - end points for D

Let start - point be ((\frac{1}{2},\frac{1}{2},0)) and end - point be ((\frac{2}{3},\frac{1}{3},\frac{2}{3})). Subtract start - point from end - point: (\frac{2}{3}-\frac{1}{2}=\frac{4 - 3}{6}=\frac{1}{6}), (\frac{1}{3}-\frac{1}{2}=\frac{2 - 3}{6}=-\frac{1}{6}), (\frac{2}{3}-0=\frac{2}{3}). Multiply by 6 to get integer indices ([1\ - 1\ 4]).

Answer:

Direction B: [2 0 0] Direction C: [0 3 1] Direction D: [1 - 1 4]