Polar coordinates: e_r and e_theta

[sara_87]

1. Homework Statement [/b]

Let e_r=(cos$$\theta$$,sin$$\theta$$) and e_theta=(-sin$$\theta$$,cos$$\theta$$).
Let P(r,$$\theta$$) be a point with e_r and e_theta at that point.
What can you say about the three quantities (e_r, e_theta and the point P) as r and $$\theta$$ vary?

Homework Equations

r: distance from origin
$$\theta$$: angle

The Attempt at a Solution

As r moves around, the e_r and e_theta change place. As r increases or decreases e_r and e_theta don't change place but as theta changes, they do change place since e_theta is always orthogonal to point P.

I feel like I'm not putting enough information. Is there something i didn't mention?
Thank you.

 

[djeitnstine]

Yes there is some information missing. You have to note that as r changes, $$\theta$$ changes with the rate that r changes, because $$e_{\theta}$$ is the derivative of $$e_{r}$$

 

[sara_87]

What do you mean, I don't understand why.

 

[djeitnstine]

Ok, starting with a (position) vector function, (I will use the cartesian coordinate system to make things appear obvious) let's say $$v=t^{2}i+2tj$$ The derivative is $$v'=2ti+2j$$. Then how does $$v$$ vary with $$v'$$? Since $$v'$$ is the derivative of $$v$$, then $$v'$$ must vary with $$v$$'s rate of change. Looking at your position functions, $$-\sin{(\theta)}$$ is the derivative of $$\cos{(\theta)}$$. Likewise for $$sin{(\theta)}$$ and $$cos{(\theta)}$$

 

[sara_87]

Oh right, i see.
Thank you