Within a cubic unit cell, sketch the following directions: (a) [101] (b) [211] (c) \([10 \overline{2}]\) (d) \([3 \overline{1} 3]\) (e) \([\overline{1} 1 \overline{1}]\) (f) \([\overline{2} 12]\) (g) \([3 \overline{1} 2]\) (h) [301]

Direction indices are written in square brackets [\(uvw\)] and represent the coordinates of the endpoint of the vector starting from origin (0,0,0) and going to the endpoint (u,v,w) in terms of the unit cell basis vectors (\(a\), \(b\), and \(c\)). If any of the coordinates are negative, they are represented as an overline.

Begin by drawing a cubic unit cell with axes \(a\), \(b\), and \(c\). Label the origin (0,0,0) and each of the eight corners of the cube with their corresponding coordinates as follows: - O: (0,0,0) - A: (1,0,0) - B: (0,1,0) - C: (0,0,1) - D: (1,1,0) - E: (1,0,1) - F: (0,1,1) - G: (1,1,1) - H: (1,1,-1)

Using the coordinates and representation of direction indices, draw the following directions within the cubic unit cell: (a) [101] - Start at the origin (0,0,0) and travel to coordinates (1,0,1). Draw a vector connecting these points. (b) [211] - Start at the origin (0,0,0) and travel to coordinates (2,1,1) within the unit cell, which is the same as coordinates (1,1,1) in the neighboring cell. Draw a vector connecting these points. (c) \([10 \overline{2}]\) - Start at the origin (0,0,0) and travel to coordinates (1,0,-2). This point lies on the line between corners H and G. Draw a vector connecting these points. (d) \([3 \overline{1} 3]\) - Start at the origin (0,0,0) and travel to coordinates (3,-1,3) within the unit cell, which is the same as coordinates (1,-1,1) in the neighboring cell. Draw a vector connecting these points. (e) \([\overline{1} 1 \overline{1}]\) - Start at the origin (0,0,0) and travel to coordinates (-1,1,-1) within the unit cell, which is the same as coordinates (0,1,0) in the neighboring cell. Draw a vector connecting these points. (f) \([\overline{2} 12]\) - Start at the origin (0,0,0) and travel to coordinates (-2,1,2) within the unit cell, which is the same as coordinates (0,1,2) in the neighboring cell. Draw a vector connecting these points. (g) \([3 \overline{1} 2]\) - Start at the origin (0,0,0) and travel to coordinates (3,-1,2) within the unit cell, which is the same as coordinates (1,-1,2) in the neighboring cell. Draw a vector connecting these points. (h) [301] - Start at the origin (0,0,0) and travel to coordinates (3,0,1). This point lies on the line between corners E and A. Draw a vector connecting these points. By following these steps, you should now have a cubic unit cell with the specified directions sketched within it.