Unit vector in polar coorindate

[athrun200]

In rectangular corr. 3i+j mean leght in x-direction =3 in y-3direction =1

However, how about in polar coorindate?
3r+1$\theta$ (r and $\theta$ are the unit verctor in polar coor., I don't know how to type it out, I hope you understand.)

Dose it mean a line with length 3 from origin and angle 1 radian from x-axis?

 

[Hootenanny]

In rectangular corr. 3i+j mean leght in x-direction =3 in y-3direction =1

However, how about in polar coorindate?
3r+1$\theta$ (r and $\theta$ are the unit verctor in polar coor., I don't know how to type it out, I hope you understand.)

Dose it mean a line with length 3 from origin and angle 1 radian from x-axis?

Yes, just as 3i+j means three "steps" in the x-direction and 1 "step" in the y-direction, 3r + 1$\theta$ means three "steps" away from the origin and 1 "step" counter-clockwise from the x-axis.

 

[athrun200]

Yes, just as 3i+j means three "steps" in the x-direction and 1 "step" in the y-direction, 3r + 1$\theta$ means three "steps" away from the origin and 1 "step" counter-clockwise from the x-axis.

How to convert the vector in retangular corrdinate to polar coordinate?
For example, 3i+j, how to write in polar corrdinate unit vector?

 

[Hootenanny]

For a detailed explanation see here: http://en.wikipedia.org/wiki/Polar_coordinate_system

In polar coordinates, the radius is simply the distance from the origin (i.e. the magnitude of your position vector). So in your example, $r = |3\hat{i} + \hat{j}| = \sqrt{10}$. The angular component is simply the angle between your vector and the positive x-direction. Since your point is in the first quadrant, this is simply $\theta=\arctan(y/x) = \arctan(1/3)$. If your point were to be in a different quadrant, you would have to adjust accordingly.

 

[athrun200]

I am still very unfamiliar with this new topic, and I still have a lot of question marks in my head.

The details of my reply please refer to my attachment

 

[Hootenanny]

I am still very unfamiliar with this new topic, and I still have a lot of question marks in my head.

The details of my reply please refer to my attachment

The unit vector $\vex{e}_\theta$ represents an "additional" rotation from the vector $\vec{e}_r$. Notice that the unit vector $\vec{e}_r$ is the direction from the origin to the point in space $\vec{r}$: http://en.wikipedia.org/wiki/File:Polar_unit_vectors.PNG.