$($a$)$ What is the unit vector pointing in the $[001]-$direction $($along a vertical side of the cube$)?$

hat$(u{)}_{[001]}=$hat$(i)+$hat$(j)+$hat$(k)$

$($b$)$ What is the unit vector pointing in the $[110]-$direction $($the diagonal of a horizontal face of the cube$)?$

hat$(u{)}_{[110]}=$

hat$(j)+$

$($c$)$ What is the unit vector pointing in the $[111]-$direction $($a diagonal of the cube$)?$

hat$(u{)}_{[111]}=$hat$(i)+$hat$(j)+$hat$(k)$

Next, think about the spherical citrus fruit. Along which of these directions do you think the spheres touch?

$($d$)$ What is the separation between the centers of the atoms at the corners of the unit cell in the hat$(z)-$direction $($along the $[001]$ direction$).$

$($e$)$ What is the separation between the centers of the atoms along the diagonal in the hat$(x)-$hat$(y)$ plane $($along the $[110]$ direction$)?$

$($f$)$ What is the sebaration between the centers of the atoms at opposite corners of the unit cell $($along the $[111]$ direction$)?$