Talha's Physics Academy
Turning Effect of Forces - Resolution of a Vector
Q.4 Define resolution of vector and derive expression for horizontal and vertical component of a vector.
Resolution of a Vector
"The process of splitting up a single vector into two or more component vectors is called the resolution of a vector."
Figure: Resolution of vector $\vec{F}$ into horizontal ($F_x$) and vertical ($F_y$) rectangular components.
Rectangular Components
Although a single vector can be resolved into an infinite number of components, it is generally most useful to resolve a vector into two components acting at right angles ($90^\circ$) to each other, known as rectangular components.
Mathematical Derivation
Suppose a vector $\vec{F}$ is represented by the line segment $OA$, which makes an angle $\theta$ with the horizontal surface $OX$. From point $A$, drop a perpendicular line $AB$ down to the horizontal axis $OX$.
- The line segment $OB$ represents the horizontal component, denoted by $F_x$.
- The line segment $AB$ represents the vertical component, denoted by $F_y$.
Consider the right-angled triangle $\triangle OAB$:
1. Horizontal Component ($F_x$)
Using the cosine ratio in $\triangle OAB$:
$\cos\theta = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{OB}{OA} = \frac{F_x}{F}$
$F_x = F \cos\theta$
2. Vertical Component ($F_y$)
Using the sine ratio in $\triangle OAB$:
$\sin\theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}} = \frac{AB}{OA} = \frac{F_y}{F}$
$F_y = F \sin\theta$
Finding the Direction of the Vector
To determine the direction ($\theta$) of the vector from its rectangular components, use the tangent ratio:
$\tan\theta = \frac{\text{Perpendicular}}{\text{Base}} = \frac{AB}{OB} = \frac{F_y}{F_x}$
$\theta = \tan^{-1}\left(\frac{F_y}{F_x}\right)$
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