Resultant of concurrent forces in space:

$Let\ the\ single\ force\ \bar R,\ be\ a\ resultant\ of\ concurrent\ force\ system\ and\ it\ acts\ through\ point\ of\ concurrency.$

$Through\ point\ A,\ the \ force\ vectors\ \bar {F_1}, \bar {F_2}, \bar {F_3}, \bar {F_4},\bar {F_5}\ pass.$

$Resultant\ force\ vector\ (\bar R)\ is\ the\ sum\ of\ all\ force\ vectors. $

$\bar R=\bar {F_1}+ \bar {F_2}+ \bar {F_3}+ \bar {F_4}+\bar {F_5}$

$\bar R= (\sum F_x)i+ (\sum F_y)j +(\sum F_z)k$

$\bar R= R_xi+ R_yj+ R_zk$

$Magnitude\ of\ resultant\ R\ is\ given\ as\ (R)=\sqrt {R^2_x+R^2_y+R^2_z}$

$Directions\ are\ given\ as: \theta_x=\cos^{-1} \bigg(\dfrac {\mathrm R_x} {\mathrm R}\bigg)$

$\theta_y=\cos^{-1} \bigg(\dfrac {\mathrm R_y} {\mathrm R}\bigg)$

$\theta_z=\cos^{-1} \bigg(\dfrac {\mathrm R_z} {\mathrm R}\bigg)$