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Callisto

  • 2 years ago

How do the plane polar coordinates work?

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  1. experimentX
    • 2 years ago
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    you know polar coordinate system right!!|dw:1349364467374:dw|

  2. Callisto
    • 2 years ago
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    Yes.. a little... But NOT in vectors!!

  3. experimentX
    • 2 years ago
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    no they are just same.|dw:1349364550790:dw|

  4. Callisto
    • 2 years ago
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    |dw:1349364634178:dw|

  5. experimentX
    • 2 years ago
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    we know that we write (x, y) as our point. In polar coordinate we write (r, theta) <--- this means this is a vector space. you have been using it unknowingly.

  6. Callisto
    • 2 years ago
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    Since it's related to sine, and cosine, it's not linear translation, right?

  7. experimentX
    • 2 years ago
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    |dw:1349364748021:dw|

  8. experimentX
    • 2 years ago
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    yep. now let's go back to unit vector for a while. in cartesian how do you represent a vector?

  9. Callisto
    • 2 years ago
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    xi + yj? i and j are unit vectors

  10. experimentX
    • 2 years ago
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    yes ... and in cartesian coordinate?

  11. Callisto
    • 2 years ago
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    Huh?! Are they different??

  12. experimentX
    • 2 years ago
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    Woops sorry ... polar coordinate??

  13. Callisto
    • 2 years ago
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    Well... Isn't it what I got stuck?

  14. experimentX
    • 2 years ago
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    \[ r \hat r + \theta \hat \theta\]

  15. experimentX
    • 2 years ago
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    now we need to find the relation between \( \hat r , \hat \theta \) and \( \hat i, \hat j \)

  16. experimentX
    • 2 years ago
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    for that we know

  17. experimentX
    • 2 years ago
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    |dw:1349365095537:dw|

  18. experimentX
    • 2 years ago
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    |dw:1349365280570:dw|

  19. experimentX
    • 2 years ago
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    |dw:1349365350758:dw|

  20. experimentX
    • 2 years ago
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    how do we define unit vectors?

  21. Callisto
    • 2 years ago
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    vector/length of vector?

  22. experimentX
    • 2 years ago
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    yep ...

  23. experimentX
    • 2 years ago
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    |dw:1349365586284:dw|

  24. experimentX
    • 2 years ago
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    can you write the relation between \( \hat r \) and i,j

  25. Callisto
    • 2 years ago
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    unit vector of r = (xi + yj) / (x^2 + y^2) ? I know it's strange...

  26. experimentX
    • 2 years ago
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    |dw:1349365738630:dw|

  27. Callisto
    • 2 years ago
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    Oh, should I express it in terms of theta?

  28. experimentX
    • 2 years ago
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    yep

  29. Callisto
    • 2 years ago
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    unit vector of r = (xi + yj) / (x^2 + y^2) \(= \frac{(rcos) \theta i +(rsin\theta)j}{r^2cos^2\theta+r^2 sin^2 \theta} \) \(= \frac{(rcos) \theta i +(rsin\theta)j}{r^2 } \) Hmm.. doesn't look good, sorry!!

  30. experimentX
    • 2 years ago
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    |dw:1349366033743:dw|

  31. experimentX
    • 2 years ago
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    can you check page no 14 again?

  32. Callisto
    • 2 years ago
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    Yes....

  33. experimentX
    • 2 years ago
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    you got your first relation right?

  34. Callisto
    • 2 years ago
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    Eh.... I was about to say no, but then, yes :|

  35. experimentX
    • 2 years ago
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    for second relation there are many tricks. this is my favourite. polar coordinate is orthogonal coordinate system. \[ \hat r \cdot \hat \theta = 0\] so change \( \theta = \theta + 90 \) you will get second relation.

  36. Callisto
    • 2 years ago
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    Hmm, sorry!! OS is not loading properly here... And... please give me sometime to work it out first.. :(

  37. experimentX
    • 2 years ago
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    sure till then i'll work out on the original method|dw:1349366641329:dw|

  38. Callisto
    • 2 years ago
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    Sorry... May I know what that e is?? I meant he wrote \(\hat e_\theta\) or \(\hat e_r\) there.

  39. experimentX
    • 2 years ago
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    |dw:1349366770653:dw|

  40. experimentX
    • 2 years ago
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    |dw:1349367409678:dw|

  41. Callisto
    • 2 years ago
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    So, they are the unit vectors?

  42. experimentX
    • 2 years ago
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    yep!!

  43. experimentX
    • 2 years ago
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    |dw:1349367720837:dw|

  44. Callisto
    • 2 years ago
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    Right, your trick works :|

  45. Callisto
    • 2 years ago
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    PQ as a vector? or...?

  46. experimentX
    • 2 years ago
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    yep. PQ

  47. experimentX
    • 2 years ago
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    |dw:1349367904332:dw|

  48. experimentX
    • 2 years ago
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    i hope you are understanding.

  49. Callisto
    • 2 years ago
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    PQ (vector) =\(r (cos(\theta +d\theta) -cos\theta )i + r (sin(\theta+d\theta) -sin\theta)j\) Sum-to-product formula?

  50. experimentX
    • 2 years ago
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    yep .. .cos C - cos D .. and ...

  51. Callisto
    • 2 years ago
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    \[PQ = r[cos(\theta+d\theta)-cos\theta] i + r[sin(\theta+d\theta)-sin\theta]j\] \[=-2rsin(\theta+\frac{d\theta}{2})sin\frac{d\theta}{2}i +2rcos(\theta+\frac{d\theta}{2})sin\frac{d\theta}{2}j\]

  52. experimentX
    • 2 years ago
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    |dw:1349368740990:dw|

  53. Callisto
    • 2 years ago
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    Isn't that |PQ| something horrible??

  54. experimentX
    • 2 years ago
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    lol ... no. Looks like i forgot cos C - cos D

  55. experimentX
    • 2 years ago
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    |dw:1349369192240:dw|

  56. Callisto
    • 2 years ago
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    Enough hint already!! (shhh!!~)

  57. experimentX
    • 2 years ago
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    |dw:1349369373117:dw|

  58. experimentX
    • 2 years ago
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    |dw:1349369423357:dw|

  59. experimentX
    • 2 years ago
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    |dw:1349369566965:dw|

  60. experimentX
    • 2 years ago
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    |dw:1349369666573:dw|

  61. experimentX
    • 2 years ago
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    **************************************************************************************************************************************************************

  62. experimentX
    • 2 years ago
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    |dw:1349369888678:dw|

  63. experimentX
    • 2 years ago
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    |dw:1349369925726:dw|

  64. experimentX
    • 2 years ago
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    similarly for theta |dw:1349370057135:dw|