Consider the previously defined coordinate systems $F_{EC}$ and $F_E.$ Denote latitude by the symbol $$ and longitude by the symbol $.$ Measure latitude positive north and negative south of the equator, $-90$deg$+90$deg. Measure longitude positive east and negative west of zero longitude, $-180$degdeg.

$($a$)$ Given the latitude ${}_E$ and longitude ${}_E$ of $F_E,$ find a transformation matrix from $F_{EC}$ to $F_E(T_{E,EC})$ that is a function only of ${}_E$ and ${}_E.$

$($b$)$ Using your answer to part $3$a$,$ transform the Earth's rotation vector in $F_{EC}$ to its representation in Earth$-$fixed coordinates at Blacksburg, Virginia, USA