Describing a position vector with polar coordinates.

[Mr Davis 97]

I have read that in polar coordinates, we can form the position vector, velocity, and acceleration, just as with Cartesian coordinates. The position vector in Cartesian coordinates is $\vec{r} = r_x \hat{i} + r_y \hat{j}$. And any choice of $r_x$ and $r_y$ maps the vector to a position in the plane. How is this done with polar coordinates? Online I have read that the position vector in polar coordinates is $\vec{r} = |r| \hat{r}$, but I don't see how this can map to any point in the plane. Don't we need an angular description as well? I don't see that in this equation.

 

[blue_leaf77]

For position vector, you can always represent a position vector as a sum between a radial vector and angular vector. But the resultant vector turns out to be another radial vector, therefore it's superfluous to use the representation which contains the angular component.

 

[Kajal Sengupta]

Polar co-ordinates for a plane involve two quantities, 'r' and 'Theta' just like 'x'and 'y' in Cartesian system . If you only specify 'r' then you are not giving the complete picture. It is like mentioning only the 'x' or 'y' in Cartesian co-ordinates.

 

[Dale]

Online I have read that the position vector in polar coordinates is ⃗r=|r|^rr→=|r|r^\vec{r} = |r| \hat{r}, but I don't see how this can map to any point in the plane.

The thing that you have to keep in mind is that in polar coordinates the basis vectors $\hat r$ and $\hat{\theta}$ are functions of the coordinates $r$ and $\theta$. So $\vec{r} = |r|\hat{r}$ should probably be written $\vec{r} = |r|\hat{r}_{(r,\theta)}$ for clarity.

 

[Chestermiller]

The thing that you have to keep in mind is that in polar coordinates the basis vectors $\hat r$ and $\hat{\theta}$ are functions of the coordinates $r$ and $\theta$. So $\vec{r} = |r|\hat{r}$ should probably be written $\vec{r} = |r|\hat{r}_{(r,\theta)}$ for clarity.

Excellent answer. The only thing I would add would be that, in polar coordinates, the two unit vectors are functions only of $\theta$ (and not r).

 

[Dale]

Excellent answer. The only thing I would add would be that, in polar coordinates, the two unit vectors are functions only of $\theta$ (and not r).

Oops, you are completely correct.

 

[Mr Davis 97]

So using the notation $\vec{r} = |r| \hat{r}_{(\theta)}$ how would I write out the vector (for example) $\vec{r} = 2\hat{i} + 4 \hat{j}?$

 

[Mark44]

So using the notation $\vec{r} = |r| \hat{r}_{(\theta)}$ how would I write out the vector (for example) $\vec{r} = 2\hat{i} + 4 \hat{j}?$

What's the angle that the vector <2, 4> makes, and what is its magnitude?