X and y components of polar unit vectors.

[jhosamelly]

Homework Statement

What are the x- and y-components of the polar unit vectors $\hat{r}$ and $\hat{\theta}$ when
a. $\theta$ = 180°
b. $\theta$ = 45°
c. $\theta$ = 215°

Homework Equations

The Attempt at a Solution

Please check if I'm correct, i'll just show my answer for a since the process is the same for a, b and c

for a.

$\hat{r_{x}}$ = r cos $\theta$
$\hat{r_{x}}$ = 1 cos 180°
$\hat{r_{x}}$ = -1

$\hat{r_{y}}$ = r sin $\theta$
$\hat{r_{y}}$ = 1 sin 180°
$\hat{r_{y}}$ = 0

in terms of theta... i don't have any idea how... please help

 

[tjackson3]

That's exactly right. If you imagine a unit circle, at 180 degrees, you're on the opposite side of the circle from 0 degrees. x = -1, y = 0.

 

[jhosamelly]

That's exactly right. If you imagine a unit circle, at 180 degrees, you're on the opposite side of the circle from 0 degrees. x = -1, y = 0.

What about theta?? who could I find its x and y component?

 

[jhosamelly]

Can someone help me how to find $\hat{\theta}$? I don't know how. thanks

 

[SammyS]

Can someone help me how to find $\hat{\theta}$? I don't know how. thanks

Do you not have a definition of the unit vector $\hat{\theta}\,?$

The unit vector $\hat{\theta}$ lies in the xy-plane and is 90° counter-clockwise from $\hat{r}\,.$

 

[jhosamelly]

Do you not have a definition of the unit vector $\hat{\theta}\,?$

The unit vector $\hat{\theta}$ lies in the xy-plane and is 90° counter-clockwise from $\hat{r}\,.$

I didn't really get what you said. sorry. can you show me an example on how to get x and y component for a then i'll do it for b and c. thanks. much appreciated.

 

[SammyS]

I didn't really get what you said. sorry. can you show me an example on how to get x and y component for a then i'll do it for b and c. thanks. much appreciated.

Well, if $(\hat{r})_x=\cos(\theta)\,,\text{ then }(\hat{\theta})_x=\cos(\theta+90^\circ)\,.$ ... etc.

Use the angle addition identity to simplify cos(θ+90°) .

 

[jhosamelly]

Well, if $(\hat{r})_x=\cos(\theta)\,,\text{ then }(\hat{\theta})_x=\cos(\theta+90^\circ)\,.$ ... etc.

Use the angle addition identity to simplify cos(θ+90°) .

so for a


$(\hat{\theta})_x=cos(180+90)$
$(\hat{\theta})_x=cos(270)$
$(\hat{\theta})_x= 0$

then

$(\hat{\theta})_y=sin (180+90)$
$(\hat{\theta})_y=sin (270)$
$(\hat{\theta})_y= -1$

am i correct?

 

[SammyS]

Yes. That's correct.