$3.$ The truss structure in Figure supports the force F at point $2.$ Analyze this structure by finite element method with two truss elements.

$3-1)$ Calculate the transformation matrix for elements $1$ and $2.$

$3-2)$ Calculate the stiffness matrix of each of the two elements based on the global coordinate system.

$3-3)$ Write the equilibrium equation for the whole truss system as below.

$[$k$]\{$q$\}=\{$f$\}$

$3-4)$ Use boundary conditions to express u$2$ and v$2$ displacement values using F$,$ L$,$ E$,$ and A$.$

$3-5)$ Use u$2$ and v$2$ obtained from $3-4$ to express F$1$x$,$ F$1$y$,$ F$3$x$,$ and F$3$y using F only.

$3-6)$ Use F$,$ L$,$ E$,$ and A to express the displacement of each node in the local coordinate system in element $1.$

$3-7)$ In element $1,$ use only F to express the strength of each node in the local coordinate system.