Obtain the relation between linear velocity and angular velocity

Consider a particle moving with uniform circular motion in anticlock wise direction with centre $O$ and radius $r$ the particle cover an arc of length $△s$ in time $△t$ moving from $A$ to $B$.
Hence the angular displacement is
$△\theta =\frac{△s}{r}$
Dividing both side by $△t$
$\frac{△\theta }{△t}=\frac{1}{r}\frac{△s}{△t}$
In the time interval $△t$ be infinitesimally small $(△t\to 0)$, then
$\underset{△t\to \theta }{\lim }\frac{△\theta }{△t}=\frac{1}{r}\left[\underset{△t\to \theta }{\lim }\frac{△s}{△t}\right]$
But $\underset{△t\to \theta }{\lim }\frac{△\theta }{△t}=\omega$ and $\underset{△t\to \theta }{\lim }\frac{△s}{△t}=V$
$\therefore \omega =\frac{V}{r}$
$V=r\omega$
In vector form, $\vec{V}=\vec{\omega }\times \vec{r}$