vectors in polar co-ordinates

**poirot** · March 12th 2011, 06:02 AM

Can anyone give me an explanation, with a simple numerical explanation of how to represent vectors as polar vectors (i'm doing it in relation velocity and acceleration vectors).

I gather that a basis for polar vectors is (cosx, sinx), (-sinx,cosx). Really I'm asking given a vector in cartesian, how do you represnt it as a lin. combination of these vectors and so write is as a polar vector. Thanks in advance

**TheEmptySet** · March 12th 2011, 07:07 AM

Originally Posted by poirot

Can anyone give me an explanation, with a simple numerical explanation of how to represent vectors as polar vectors (i'm doing it in relation velocity and acceleration vectors).

I gather that a basis for polar vectors is (cosx, sinx), (-sinx,cosx). Really I'm asking given a vector in cartesian, how do you represnt it as a lin. combination of these vectors and so write is as a polar vector. Thanks in advance

So you have your two polar basis vectors

$\mathbf{\hat{r}}=\cos(\theta)\vec{i}+\sin(\theta)\ vec{j}$ and

$\mathbf{\hat{\theta}}=-\sin(\theta)\vec{i}+\cos(\theta)\vec{j}$

Now if you have a vector and since we know that r hat and theta hat are an orthonormal basis we can just project a new vector onto them

Given
$\vec{v}(x,y)=x\vec{i}+y\vec{j}$

$\vec{v}(r,\theta)=[\vec{v}(r\cos(\theta),r\sin(\theta))\cdot \mathbf{\hat{r}}]\mathbf{\hat{r}}+[\vec{v}(r\cos(\theta),r\sin(\theta))\cdot \mathbf{\hat{\theta}}]\mathbf{\hat{\theta}}$

$\vec{v}=(r\cos^2(\theta)+r\sin^2(\theta))\mathbf{\ hat{r}}+(-r\sin(\theta)\cos(\theta)+r\sin(\theta)\cos(\theta ))\mathbf{\hat{\theta}}=r\mathbf{\hat{r}}$

**poirot** · March 12th 2011, 11:03 AM

Originally Posted by TheEmptySet

So you have your two polar basis vectors

$\mathbf{\hat{r}}=\cos(\theta)\vec{i}+\sin(\theta)\ vec{j}$ and

$\mathbf{\hat{\theta}}=-\sin(\theta)\vec{i}+\cos(\theta)\vec{j}$

Now if you have a vector and since we know that r hat and theta hat are an orthonormal basis we can just project a new vector onto them

Given
$\vec{v}(x,y)=x\vec{i}+y\vec{j}$

$\vec{v}(r,\theta)=[\vec{v}(r\cos(\theta),r\sin(\theta))\cdot \mathbf{\hat{r}}]\mathbf{\hat{r}}+[\vec{v}(r\cos(\theta),r\sin(\theta))\cdot \mathbf{\hat{\theta}}]\mathbf{\hat{\theta}}$

$\vec{v}=(r\cos^2(\theta)+r\sin^2(\theta))\mathbf{\ hat{r}}+(-r\sin(\theta)\cos(\theta)+r\sin(\theta)\cos(\theta ))\mathbf{\hat{\theta}}=r\mathbf{\hat{r}}$

why, in writing a polar vector with respect to the basis, have you used rcosx and rsinx when your input was r and x? I'm using x for theta btw.

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