Write all the unit vectors in XY-plane.

Solution

Let $straight r with rightwards arrow on top space equals space straight x space straight i with hat on top space plus space space straight y space straight j with hat on top space$ be a unit vector $OP with rightwards arrow on top$ in XY - plane.
$therefore space space space space space space OP space equals open vertical bar straight r with rightwards arrow on top close vertical bar space equals space 1 straight P space is space left parenthesis straight x comma space straight y right parenthesis$
From P draw $PM space perpendicular space OX.$
In rt. angled $increment OMP comma$
$straight x over 1 space equals space cos space straight theta comma space space space straight y over 1 space equals space sin space straight theta$

$therefore space straight x space equals space cos space straight theta comma space space space straight y space equals space sin space straight theta therefore space space straight r with rightwards arrow on top space equals space left parenthesis cos space straight theta right parenthesis space straight i with hat on top space plus space left parenthesis sin space straight theta right parenthesis space straight j with hat on top space space space... left parenthesis 1 right parenthesis Here space open vertical bar straight r with rightwards arrow on top close vertical bar space equals space square root of cos squared straight theta plus space sin squared straight theta end root space equals square root of 1 space equals space 1$
Also, as $straight theta$ varies from 0 to $straight pi$ , the point P traces the circle r² + y² = 1 counterclockwise, and this covers all possible directions.
∴ (1) gives every unit vector in the XY-plane.