Problem #$1$: $(40$ points$)$ A mass $m=50kg$ is suspended at point $A,$ which is supported by cables

$AB,AC,$ and $AD,$ as shown in Figure $1.$ An $x-y-z$ coordinate system is drawn on the figure. Point

$A$ is located on the $y-$axis at a vertical distance of $1\mathrm{.}5$ m above the horizontal $x-z$ plane. The

height of each post $,$ and $D$ is $2$ m $.$ The locations of points $B,C,$ and $D$ can be determined

by the dimensions on the figure and the height of the posts. $($Note: Point $C$ lies in the $y-z$ plane;

and point $D$ lies in the $x-y$ plane.$)$ The resulting tensions in cables $AB,AC,$ and $AD$ are $T_{AB},T_{AC},$

and $T_{AD},$ respectively.

a$)$ Find the unit vector $u_{AB}($directed from point $A$ to point $B),$ the unit vector $u_{AC}($directed

from point $A$ to point $C),$ and the unit vector $u_{AD}($directed from point $A$ to point $D).$ Express

the results in terms of the unit vectors $i,j,$ and $k.$

b$)$ Express the force $T_{AB}$ in cable $AB,$ the force $T_{AC}$ in cable $AC,$ and the force $T_{AD}$ in cable $AD$

in terms of the tensions $T_{AB},T_{AC},$ and $T_{AD}$ and the unit vectors $i,j,$ and $k.$

c$)$ Without solving them, find the three scalar equations that determine the tensions $T_{AB},T_{AC},$

and $T_{AD}.$

d$)$ Calculate the tensions $T_{AB},T_{AC},$ and $T_{AD}$ in cables $AB,AC,$ and $AD,$ respectively.