**Briefly explain the determination of a force from its perpendicular components?**

Difficulty: Hard

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

Consider **F _{X}** and

**F**as the perpendicular components of a force

_{y}**F**. These perpendicular components

**F**and

_{X}**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**respectively. That is

_{y}**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

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

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

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