A particle's displacement is expressed by r = A( i ω t + j ω t) ....

Given the position vector of the particle as $\vec{r} = A(\cos\omega t \, \hat{i} + \sin\omega t \, \hat{j})$.

The coordinates of the particle at any time $t$ are given by:
$x = A \cos\omega t$
$y = A \sin\omega t$

Squaring and adding both equations, we get:
$x^2 + y^2 = A^2 (\cos^2\omega t + \sin^2\omega t)$

$x^2 + y^2 = A^2$

This is the equation of a circle with radius $A$ and center at the origin. Therefore, the particle executes motion on a circle.