$2.)$ The cross section of a solid beam is shown in the figure viewed opposite axis $x.$ The absolute values of the stresses arising in point $P$ from the $V$ and $M$ internal forces are given. $($Point $P$ is located on the contour of the cross$-$section, under the centroid.$)$

$|{}_x^P|=35$MPa

$|{}_x^P|=40$MPa

a$)$ Determine the sign of the normal stress and the direction of the shear stress at point $P!$ What is the stress state of point $P?$

b$)$ Assign a right$-$handed coordinate system $(u,v,x)$ to point $P$ where $v-x$ is the plane of stresses. Indicate axes precisely.

c$)$ Find the magnitude and direction of principal stresses at point $P($estimate directions of principal stresses by superposition of effects of normal and shearing stresses$).$ Display your results in a sketch viewed opposite $u$ with axes $v$ and $x$ directed down and left, respectively.

d$)$ What is the mechanical and what is the mathematical meaning of the first prinipal direction of stresses?