# Why are vertical and horizontal motion considered independent?

##### 1 Answer
Nov 2, 2016

We know that in motion problems, displacement, velocity and acceleration all are vector quantities. Only for the sake of simplicity these are written as $s , v \mathmr{and} a$.
For sake of completeness these should be written as $\vec{s} , \vec{v} \mathmr{and}$$\vec{a} .$

We also know that for vector quantities, orthogonal component is always equal to zero as shown below.

For any vector $\vec{X}$, component in the direction which makes an angle $\theta$ with the vector is given by $\vec{X} \cos \theta$.
It follows that for $\theta = {90}^{\circ}$, (i.e., in the perpendicular direction)
$\cos {90}^{\circ} = 0$
$\implies \vec{X} \cos {90}^{\circ} = 0$

As such vertical and horizontal components of motion can be treated independently.