# Describing a position vector with polar coordinates.

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## Main Question or Discussion Point

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
Homework Helper
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.

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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
Mentor
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.

Molar
Chestermiller
Mentor
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 and Molar
Dale
Mentor
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.

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
Mentor
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?