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! :)

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.

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.

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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! :)

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?

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 :)