Figure $1$

As shown in Figure $1,$ a box weighing $100$ lb is suspended by a series of cables affixed to a wall at $\backslash ($ B $\backslash ),$ to the ceiling at $\backslash ($ D $\backslash ),$ and to the ground at $\backslash ($ E $\backslash )$ and $\backslash ($ F $\backslash ).$

$($a$)$ Express position vectors $\backslash (\backslash$mathbf$\{$r$\}\_\{\backslash$mathrm$\{$AB$\}\},\backslash$mathbf$\{$r$\}\_\{\backslash$mathrm$\{$AC$\}\},\backslash$mathbf$\{$r$\}\_\{\backslash$mathrm$\{$FC$\}\}\backslash )$ and $\backslash (\backslash$mathbf$\{$r$\}\_\{\backslash$mathrm$\{$FA$\}\}\backslash )$ in Cartesian vector form.

$($b$)$ Determine the tension in cables $\backslash ($ A B $\backslash )$ and $\backslash ($ A C $\backslash )$ and express as Cartesian vectors pulling on $\backslash ($ A $\backslash ).$

$($c$)$ By using appropriate dot$-$product relationships, determine the angle between cables $\backslash ($ C F $\backslash )$ and $\backslash ($ C A $\backslash ).$

$($d$)$ Determine the moment of the force $\backslash (\backslash$mathrm$\{$F$\}\_\{\backslash$mathrm$\{$AB$\}\}\backslash )$ about cable $\backslash ($ C F $\backslash )$ and express it in Cartesian vector form.

$($e$)$ Express the moment determined in Part $($d$)$ in terms of magnitude and coordinate angles.

$[3$ marks$]$