In order to analyze the structure shown below, it is necessary to model each joint as a particle in equilibrium. Analyze joint $A$ as a particle with four forces acting upon it: the force $P=500lb,$ the compressive forces $F_{AB}$ and $F_{AD}($forces point toward $A),$ and the tensile force $F_{AC}($force points away from $A).$ Use $=30$ and $=40.$ The direction angles for bar $AD,$ as measured from point $A,$ are $=105$ and $=90.$ The force $P$ acts in the $y-$direction.

A$)$ Determine the direction angle $$ for bar $AD($as measured from point A$).$

B$)$ Write the forces acting at point $A$ in Cartesian vector form $($you must simplify$).$

C$)$ Determine the forces in bars AB$,$ AC$,$ and AD$.$

D$)$ Determine the length of bar $AD$ and the coordinates of point $D.$

E$)$ Determine the magnitude of the component of force $P$ projected perpendicular to plane ABD.