Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.

Solution

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

$straight x over 1 space equals space cos space 30 degree space space space or space space space straight x space equals space fraction numerator square root of 3 over denominator 2 end fraction$
and $straight y over 1 space equals space sin space 30 degree space space space space or space space space straight y space equals space 1 half$
$therefore space space space space straight r with rightwards arrow on top space equals space fraction numerator square root of 3 over denominator 2 end fraction straight i with hat on top space space plus space 1 half straight j with hat on top.$