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Vector Rotation About Arbitrary Axis

  1. Dec 11, 2015 #1
    I am new to this forum. I was reading this document :
    http://math.kennesaw.edu/~plaval/math4490/rotgen.pdf

    Here the author says that from this figure
    http://i.stack.imgur.com/KBw9l.png

    that we can express $v_{\perp}$ like this :
    $$T (v_{\perp}) = \cos(\theta) v_{\perp} + \sin(\theta) w$$

    I don't understand this part. Can anybody explain how $$T(v_{\perp}) $$ is $$\cos(\theta) v_{\perp} + \sin(\theta) w$$?
     
  2. jcsd
  3. Dec 12, 2015 #2
  4. Dec 12, 2015 #3

    Erland

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    Isn't this obvious from the figure? I see no problem at all.

    Although what should be a circle looks like an ellipse in the figure I see on my computer. Could be something wrong with the graphics the author used. Anyway, if you also see an ellipse, think of it as a circle!
     
  5. Dec 12, 2015 #4
    Why we are experessing the Vector as addition of cos(θ)v⊥ and sin(θ)w ? Is it the result of 2 vector addition like Vab = Va + Vb?
     
  6. Dec 12, 2015 #5

    Erland

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    Is your question why it is correct to do this, or what is the use of doing this?
     
  7. Dec 12, 2015 #6
    why it is correct to do this ?
     
  8. Dec 12, 2015 #7

    Erland

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    Don't you know how to add vectors geometrically, with the parallellgram law?

    Here we are studying a linear tranformation which rotates every vector in the plane perpendicular to the rotation axis by the angle ##\theta## about that axis. Then ##v_\perp## is mapped to ##T(v_\perp)## in the figure. And ##v_\perp## and ##w## are perpendicular unit vectors in this plane, so we obtain the relation stated.
     
  9. Dec 12, 2015 #8
    Thanks now i understood. Sorry sometimes i am dumb and gets confused with simpler things.
     
  10. Dec 12, 2015 #9

    Erland

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    No problem, you're very welcome! We all get confused sometimes.
     
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