$(120$ pts$.)$ The $3$D stress state of a point in a steel machine component is physically determined using piezoelectric strain gauges, Wheatstone Bridge, and generalized Hooke's Law to be as follows:

${}_{}=-119\mathrm{.}4$MPa,${}_{yy}=103\mathrm{.}5$MPa,${}_z=-87\mathrm{.}6$MPa,

${}_{xy}=-5\mathrm{.}92$MPa,${}_{yz}=-7\mathrm{.}96$MPa,${}_{xz}=11\mathrm{.}94$MPa

Determine

$($a$)(60-$pt$)$ all the $3$ principal stresses

$($b$)(30-$pt$)$ the orientation of the maximum principal stress by giving the corresponding unit normal vector

$($c$)-$pt$)$ the magnitude and direction of ${}_{\text{m}\text{a}\text{x}}$

$($d$)(10-$pt$)$ the direction along which Mode I fracture is most likely to occur subject to either known or unknown stress concentration?