$2\mathrm{.}3-2.$ Determine the principal stresses associated with the following states of stress $($in MPa $).$ Also, for each state, verify that Eqs. $2\mathrm{.}3-2$ and $2\mathrm{.}3-3$ do indeed provide the same numerical values of $I_2$ and $I_3.$

$($b$){}_x=-80,{}_y=40,{}_z=-40,{}_{xy}=-40,{}_{yz}=120,{}_{zx}=80$

$2\mathrm{.}3-3.$ For the states of stress in Problem $2\mathrm{.}3-2,$ determine the direction cosines of the maximum principal stress.

$I_1={}_x+{}_y+{}_z$

$I_2={}_x{}_y+{}_y{}_z+{}_z{}_x-{}_{xy}^2-{}_{yz}^2-{}_{zx}^2$

$I_3={}_x{}_y{}_z+2{}_{xy}{}_{yz}{}_{zx}-{}_x{}_{yz}^2-{}_y{}_{zx}^2-{}_z{}_{xy}^2$

$I_1={}_1+{}_2+{}_3$

$I_2={}_1{}_2+{}_2{}_3+{}_3{}_1$

$I_3={}_1{}_2{}_3$