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 x2 + y2 = 1 counterclockwise, and this covers all possible directions.
i,e $\vec{r}=\cos \theta \hat{i}+\sin \theta \hat{j} ; \quad \theta \in(0,2 \pi)$, 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 x2 + y2 = 1 counterclockwise, and this covers all possible directions.
i,e $\vec{r}=\cos \theta \hat{i}+\sin \theta \hat{j} ; \quad \theta \in(0,2 \pi)$, represents all the unit vectors in a plane.
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