The stress components in a Cartesian coordinate system at a point $P$ are given by

$({}_{ij})=([3,2,2],[2,4,0],[2,0,2])$

Find:

a$)$ The traction at $P$ on a plane normal to the $x_1$ axis.

b$)$ The traction at $P$ on the plane whose normal has direction ratios $1$:$-3$:$2$

c$)$ The traction at $P$ on a plane through $P$ parallel to the plane $x_1+2x_2+3x_3=1$

d$)$ The principal stress components at $P$

e$)$ The directions of the principal axes of stress at $P.$ Verify that they are mutually orthogonal.