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I Describing a position vector with polar coordinates.

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  1. Aug 30, 2016 #1
    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.
     
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  3. Aug 30, 2016 #2

    blue_leaf77

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    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.
     
    Last edited: Aug 30, 2016
  4. Aug 30, 2016 #3
    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.
     
  5. Aug 30, 2016 #4

    Dale

    Staff: 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.
     
  6. Aug 30, 2016 #5
    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).
     
  7. Aug 30, 2016 #6

    Dale

    Staff: Mentor

    Oops, you are completely correct.
     
  8. Aug 30, 2016 #7
    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}?##
     
  9. Aug 30, 2016 #8

    Mark44

    Staff: Mentor

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