$(25$ pts$)($a$)$ Assuming the unit cells in the lattice shown as cubic, mark lattice points $211,012,102,\frac{1}{2}\frac{1}{2}2,\frac{1}{2}02,$ and $\frac{1}{2}\frac{1}{2}2$ by placing filled circles. $($b$)$ Write the three indices for the two lattice points marked with open circles. $($c$)$ Write the directions for the two rays indicated by the two arrows. $($d$)$ Find the angle $$ between these two directions. $($e$)$ determine the Miller indices for the shadecholane.