Determination of a Force or a vector from its Perpendicular Components:

Consider F_X and F_y as the perpendicular components of a force F. These perpendicular components F_X and F_y

 are represented by lines OP and PR respectively.

According to head to the tail rule:

 OR = OP + PR

Thus, OR will completely represent the force F whose x and y-components 

are F_X and F_yrespectively. That is

F = F_X + F_y

The magnitude of resultant force/Magnitude of resultant vector:

The magnitude of the force F can be determined using the right-angled triangle OPR

As

$\left(OR\right)^{2}=\left(OP\right)^{2}+\left(PR\right)^{2}$

$F^{2}=Fx^{2}+Fy^{2}$

Hence

F= $\surd Fx^{2}+Fy ^{2}$ (i)

Direction of the resultant force/Direction of the resultant vector:

The direction of the force F with x-axis is given by

$\tan\theta=\frac{PR}{OP}=\frac{Fy}{Fx}$

$\theta=\tan^{-1}\frac{Fy}{Fx}$