$1\mathrm{.}2$ Consider an $x^{\text{'}}-y^{\text{'}}-z^{\text{'}}$ coordinate system whose basis vectors $e_1^{\text{'}},e_2^{\text{'}}$ and $e_3^{\text{'}}$ are oriented in the di$-$

rections $\{\frac{1}{\sqrt[\phantom{2}]{2}},\frac{1}{\sqrt[\phantom{2}]{2}},0\},\{-\frac{1}{2},\frac{1}{2},\frac{1}{\sqrt[\phantom{2}]{2}}\}$ and $\{\frac{1}{2},-\frac{1}{2},\frac{1}{\sqrt[\phantom{2}]{2}}\},$ respectively, relative to the $x-y-z$

coordinate system.

$($a$)$ Determine the strain transformation matrix $T_{}.$

$($b$)$ If at a point in a composite structure the strains in the $x-y-z$ coordinate system are

${}_x=1000,{}_y=-1500,{}_z=500,$

${}_{yz}=100rad,{}_{xz}=300rad,{}_{xy}=800rad$

determine the strains in the $x^{\text{'}}-y^{\text{'}}-z^{\text{'}}$ coordinate using the strain transformation matrix $T_{}.$