Consider the $Al{F}_3$ molecule

$($a$)$ Apply the projection operator method to derive the symmetry coordinates for $Al{F}_2.$

$($b$)$ Determine the symmetries of the vibrational modes.

$($c$)$ Draw pictures of each vibrational mode and identify any rotational modes

$($d$)$ Use the $C_3$ character table below to determine the symmetry doordinates.

$\backslash$table$[[D_{3h},$E$,2C_3(z),3C_2,{}_h(xy),2S_3,3{}_v,\backslash$table$[[\backslash$table$[[$linear functions,$],[$rotations$]]]],\backslash$table$[[\backslash$table$[[$quadratic$],[$functions$]]]],\backslash$table$[[\backslash$table$[[$cubic$],[$functions$]]]]],[A_1^{\text{'}},+1,+1,+1,+1,+1,+1,-,x^2+y^2,z^2,x(x^2-3y^2)$