Write all the unit vectors in the $XY$-plane.

(N/A) Let $\vec{r} = x \hat{i} + y \hat{j}$ be a unit vector in the $XY$-plane.
Since $\vec{r}$ is a unit vector,its magnitude is $|\vec{r}| = 1$,which implies $\sqrt{x^2 + y^2} = 1$,or $x^2 + y^2 = 1$.
We can represent any point on the unit circle in the $XY$-plane using the parameter $\theta$,where $\theta \in [0, 2\pi)$.
Thus,we can set $x = \cos \theta$ and $y = \sin \theta$.
Substituting these into the expression for $\vec{r}$,we get:
$\vec{r} = \cos \theta \hat{i} + \sin \theta \hat{j}$
where $\theta$ is the angle the vector makes with the positive $X$-axis.
As $\theta$ varies from $0$ to $2\pi$,this expression generates all possible unit vectors in the $XY$-plane.