A three$-$legged structure is shown below. It is attached to the ground at points B

$=(3\mathrm{.}03$

$,0$

$,0)$

and C

$=(3\mathrm{.}03$

$,0$

$,0)$

and connected to the ceiling at point A

$=(4\mathrm{.}31$

$,4\mathrm{.}66$

$,4\mathrm{.}07)$

$.$ The three legs connect at point D

$=(0$

$,3\mathrm{.}77,4\mathrm{.}00)$

$,$ where two forces, F

and P

$,$ act. Force F

is given by F$=6\mathrm{.}40$ i$20\mathrm{.}0$ j$+30\mathrm{.}0$ k N

; P

has magnitude $45\mathrm{.}0$ N

and direction angles $\backslash$alpha $=125\mathrm{.}0$

$,\backslash$beta $=54\mathrm{.}5$

$,$ and $\backslash$gamma $=54\mathrm{.}7$

for the x$,$ y$,$ and z axes, respectively. Part A $-$ Finding the Cartesian components of a force described by direction angles

Find the Cartesian components of force P

acting in the x$,$ y$,$ and z directions given P$=45\mathrm{.}0$ N

$,\backslash$alpha $=125\mathrm{.}0$

$,\backslash$beta $=54\mathrm{.}5$

$,$ and $\backslash$gamma $=54\mathrm{.}7$

$.$ Recall that $\backslash$alpha

is the angle between the vector and the x axis, $\backslash$beta

is the angle between the vector and the y axis, and $\backslash$gamma

is the angle between the vector and the z axis.

Express your answers, separated by commas, to three significant figures. Part B $-$ Finding the angle between forces

Find the angle $\backslash$theta

between forces F

and P

$.$

Express your answer to three significant figures in degrees. Part C $-$ Determining the force along a member

Determine the magnitude, FDA

$,$ acting along member DA

$,$ due to the applied force F$=6\mathrm{.}40$ i$20\mathrm{.}0$ j$+30\mathrm{.}0$ k N

$.$

Express your answer to three significant figures and include the appropriate units. Part D $-$ Finding the component of a force perpendicular to a direction

Given F$=6\mathrm{.}40$ i $20\mathrm{.}0$ j $+30\mathrm{.}0$ kN

$,$ find the component of F

that acts perpendicular to member DA

such that the vector addition of the perpendicular and parallel components of F $($F$=$F$+$F$\backslash |)$

with respect to DA equals F

$.$ Express your answer in component form.

Express your answers, separated by commas, to three significant figures.

$$