# Derive the relationship between Unit Polar Basis and Cartesian Basis

• Aug 28th 2011, 10:25 AM
divinelogos
Derive the relationship between Unit Polar Basis and Cartesian Basis
Question: Use the triangle law of vector addition to derive the following relationships between the unit polar basis and the standard cartesian basis.

Er=cos(theta)i+sin(theta)j

i=cos(theta)er-sin(theta)e(theta)

(there's two other relations but you know what they are, it would get too messy to write them)

I have drawn some diagrams but I'm not sure how to write the proof exactly...

Any ideas would be appreciated! :)
• Aug 28th 2011, 11:00 AM
Plato
Re: Derive the relationship between Unit Polar Basis and Cartesian Basis
Quote:

Originally Posted by divinelogos
Question: Use the triangle law of vector addition to derive the following relationships between the unit polar basis and the standard cartesian basis.
Er=cos(theta)i+sin(theta)j
i=cos(theta)er-sin(theta)e(theta)

I honestly do not understand what that question is asking.
I cannot think what "Use the triangle law of vector addition to derive..." could mean.
What does "the unit polar basis" mean?
Would you try to rewrite this question? Or at least explain what it means.
• Aug 28th 2011, 10:26 PM
divinelogos
Re: Derive the relationship between Unit Polar Basis and Cartesian Basis
Hey Plato :)

Here's a picture of the problem; it might clarify what is meant by the unit polar basis. It's problem number 6.

ImageShack&#174; - Online Photo and Video Hosting

I would assume it means a 2 dimensional basis where the first vector is "r" and the second "theta", and each is a unit vector.

Thanks for any help! :)
• Aug 29th 2011, 03:37 AM
HallsofIvy
Re: Derive the relationship between Unit Polar Basis and Cartesian Basis
this doesn't quite make sense: 'first vector is "r" and the second "theta"' because those are numbers, not vectors. I think what you mean rather is that we have different basis vectors at each point. One of the basis vectors, $\displaystyle \vec{v_r}$, at (x, y), is the unit vector whose direction is away from the origin, along the line y= x, and the other, $\displaystyle \vec{v_\theta}$ is the unit vector perpendicular to that in the direction of increasing angle. Those. I presume, are your "er" and "e(theta)", respectively.

But I don't know what you mean by "Er" and "i"? did you mean "er" again? And is "i" the unit vector in the x-direction?
• Aug 30th 2011, 04:48 PM
divinelogos
Re: Derive the relationship between Unit Polar Basis and Cartesian Basis
Quote:

Originally Posted by HallsofIvy
this doesn't quite make sense: 'first vector is "r" and the second "theta"' because those are numbers, not vectors. I think what you mean rather is that we have different basis vectors at each point. One of the basis vectors, $\displaystyle \vec{v_r}$, at (x, y), is the unit vector whose direction is away from the origin, along the line y= x, and the other, $\displaystyle \vec{v_\theta}$ is the unit vector perpendicular to that in the direction of increasing angle. Those. I presume, are your "er" and "e(theta)", respectively.

But I don't know what you mean by "Er" and "i"? did you mean "er" again? And is "i" the unit vector in the x-direction?

It's easier to understand if you look at the picture (posted above). I didn't know how to type that up :)