Question
Write all the unit vectors in $XY-$plane.
✓
Answer
Let $\vec{r}=x \hat{i}+y \hat{j}$ be a unit vector in $XY-$plane.
Then, from the figure below, we have $x = \cos \theta$ and $y = \sin \theta ($since $|\vec r| = 1).$
So, we may write the vector $\vec r$ as
$\vec{r}(=\vec{\mathrm{OP}})=\cos \theta ~ \hat{i}+\sin \theta ~\hat{j} .........(i)$
Clearly, $|\vec{r}|=\sqrt{\cos ^{2} \theta+\sin ^{2} \theta}=1$
Also, as $\theta$ varies from $0 ~to ~2\pi$,
the point $P$ traces the circle $x^2 + y^2 = 1$ counterclockwise, and this covers all possible directions.
i,e $\vec{r}=\cos \theta \hat{i}+\sin \theta \hat{j} ; \theta \in(0,2 \pi)$, represents all the unit vectors in a plane.
Then, from the figure below, we have $x = \cos \theta$ and $y = \sin \theta ($since $|\vec r| = 1).$
So, we may write the vector $\vec r$ as
$\vec{r}(=\vec{\mathrm{OP}})=\cos \theta ~ \hat{i}+\sin \theta ~\hat{j} .........(i)$
Clearly, $|\vec{r}|=\sqrt{\cos ^{2} \theta+\sin ^{2} \theta}=1$
Also, as $\theta$ varies from $0 ~to ~2\pi$,
the point $P$ traces the circle $x^2 + y^2 = 1$ counterclockwise, and this covers all possible directions.
i,e $\vec{r}=\cos \theta \hat{i}+\sin \theta \hat{j} ; \theta \in(0,2 \pi)$, represents all the unit vectors in a plane.
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