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

$=(3\mathrm{.}61$

$,0$

$,0)$

and C

$=(3\mathrm{.}61$

$,0$

$,0)$

and connected to the ceiling at point A

$=(5\mathrm{.}11$

$,3\mathrm{.}90$

$,4\mathrm{.}08)$

$.$ The three legs connect at point D

$=(0$

$,3\mathrm{.}32,2\mathrm{.}50)$

$,$ where two forces, F

and P

$,$ act. Force F

is given by F$=6\mathrm{.}00$ i$10\mathrm{.}0$ j$+21\mathrm{.}6$ k N

; P

has magnitude $25\mathrm{.}0$ N

and direction angles $\backslash$alpha $=139\mathrm{.}0$

$,\backslash$beta $=68\mathrm{.}0$

$,$ and $\backslash$gamma $=57\mathrm{.}4$

for the x$,$ y$,$ and z axes, respectively.$($Figure $1)$Given F$=6\mathrm{.}00$ i $10\mathrm{.}0$ j $+21\mathrm{.}6$ 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.