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

Explanation

Let, in XY-plane, OP makes `30^(@)` angle from X-axis, `60^(@)` angle from Y-axis and `90^(@)` from Z-axis. Therefore, direction cosines of vector OP are cos `30^(@)`, cos `60^(@)` and `90^(@)` i.e., `(sqrt(3))/(2),(1)/(2),0`.
Therefore, OP `=(sqrt(3))/(2)hat(i)+(1)/(2)hat(j)`
`|vec(OP)|-sqrt(((sqrt(3))/(2))^(2)+((1)/(2))^(2))`
`= sqrt((3)/(4)+(1)/(4))=sqrt(1)=1`
Therefore required unit vector in XY-plane is `(sqrt(3))/(2)hat(i) + (1)/(2)hat(j)`.