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Rotation Matricies

  1. Oct 26, 2009 #1
    1. The problem statement, all variables and given/known data
    Hey guys, I'm not sure if this bit is relevant but the first part of the question is... 'The diagram shows a triangle with vertices O, A(1,2) , B (0,2). The question I need help with is 'Each of the following matricies represents a rotation about the origin, Find the angle and direction of rotation in each case'

    (1/2 -[tex]\sqrt{3}[/tex]/2)
    ([tex]\sqrt{3}[/tex]/2 1/2)

    Imagine its one big set of brackets around the above matrix not 2 smaller ones :)

    2. Relevant equations
    I'm also not sure whether these are of relivance but x' = ax + cy and y' = bx + dy
    also I have the matrix
    cos[tex]\theta[/tex] -sin[tex]\theta[/tex]
    sin[tex]\theta[/tex] cos[tex]\theta[/tex]


    3. The attempt at a solution
    I have been messing about with this for a while attemting to sub in co ordinates to the x' equations and y' equations but I feel I am getting nowhere. I would appreciate any help.
    Thanks in advance guys
    Breen155
     
  2. jcsd
  3. Oct 26, 2009 #2
    [tex]\begin{pmatrix} a & -b \\ b & a \end{pmatrix} = \begin{pmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end{pmatrix}[/tex]

    [tex] a = \frac{1}{2}[/tex]
    [tex] b = \frac{\sqrt{3}}{2}[/tex]


    Can you use the definition of equality of matrices to get two equations for theta?
     
    Last edited: Oct 26, 2009
  4. Oct 26, 2009 #3
    not too sure what you mean sorry I only started teaching this to myself this morning :S erm... is it costheta - sin theta = .5 and sintheta + costheta = sqt3/2 ?
     
  5. Oct 26, 2009 #4
    I slipped up on my notation. I fixed my last post. Try to get the equations again because they aren't right. Remember, get them by using the definition of equality for matrices.
     
  6. Oct 26, 2009 #5
    costheta = .5 and sintheta = sqrt3/2 ? :)
     
  7. Oct 26, 2009 #6
    correct. Now you just need to find theta.
     
  8. Oct 26, 2009 #7
    theta is 60 degrees but how do i tell the direction of rotation, clockwish or anticlockwise ? :) (thanks for the help so far btw)
     
  9. Oct 26, 2009 #8
    Remember that a positive theta corresponds, by convention, to a counterclockwise rotation. If it were a clockwise rotation of 60 degrees, then theta would be -60 degrees. This is equivalent to a counterclockwise rotation of 300 degrees, and you see that the sine and cosine of 300 degrees is exactly equal to the sine and cosine of -60 degrees. Of course, this means the question is a bit vague, since you can change the direction of rotation just by changing the angle. If you have both a direction with a correct angle, however, it should be fine.
     
  10. Oct 26, 2009 #9
    Ok I understand now. Thanks for all the help guys
     
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