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Math Help - Transforation geometry -- using matrices!

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    Transforation geometry -- using matrices!

    This is a question from GCSE level, so please offer a simple explanation if possible.

    I have been given the following:

    To rotate a point (x,y) 90deg anti-clockwise, pre-multiply the x y matrix with:

    Code:
    ( 0 -1 )
      1  0
    Similarly, for a an anti-clockwise rotation of 180deg, it's:
    Code:
    -1  0
    0   -1
    p.s., how can I create a matrix in this forum in the math tag?

    It's tough to remember so memorize so many matrices when there are very similar looking matrices for reflection (on x axis, y axis and y = -x). I think perhaps, an explanation of how these matrices are created will help me. Or maybe some pointers on how to memorize at least 8-9 such matrices.

    Thanks ...
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    Quote Originally Posted by struck View Post
    This is a question from GCSE level, so please offer a simple explanation if possible.

    I have been given the following:

    To rotate a point (x,y) 90deg anti-clockwise, pre-multiply the x y matrix with:

    Code:
    ( 0 -1 )
      1  0
    Similarly, for a an anti-clockwise rotation of 180deg, it's:
    Code:
    -1  0
    0   -1
    p.s., how can I create a matrix in this forum in the math tag?

    It's tough to remember so memorize so many matrices when there are very similar looking matrices for reflection (on x axis, y axis and y = -x). I think perhaps, an explanation of how these matrices are created will help me. Or maybe some pointers on how to memorize at least 8-9 such matrices.

    Thanks ...
    The matrix that rotates by an anti-clockwise angle of \, \theta \, is

    \left( \begin{array}{cc}<br />
\cos \theta & -\sin \theta \\<br />
\sin \theta & \cos \theta \end{array} \right)


    p.s. The latex code I used for generating this matrix is

    [tex]\left( \begin{array}{cc}
    \cos \theta & -\sin \theta \\
    \sin \theta & \cos \theta \end{array} \right)[/tex]
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  3. #3
    max
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    I have found plenty to help me with counter clockwise rotations but when its clockwise the values i get for theta when i do arc cos and arc sin are both different, can you tell me why please?

    Max
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    Quote Originally Posted by max View Post
    I have found plenty to help me with counter clockwise rotations but when its clockwise the values i get for theta when i do arc cos and arc sin are both different, can you tell me why please?

    Max
    Give an example.

    To rotate clockwise, all you do is replace \theta with - \theta in the rotation matrix.
    Last edited by mr fantastic; March 18th 2008 at 05:08 AM. Reason: Fixed latex and spelling
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  5. #5
    max
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    Find theta and hence the angle of rotation and the direction of

    top row of matrix (-1/2) (1/2)
    bottom row (-1/2) (-1/2)

    Clockwise direction but
    when you calculate arccos (-1/2) you get 135 deg which is correct but
    arcsin (-1/2) is -45 deg

    What am I missing?

    Max

    matrix attached
    Attached Thumbnails Attached Thumbnails Transforation geometry -- using matrices!-matrix.jpg  
    Last edited by max; March 18th 2008 at 04:23 AM.
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    Quote Originally Posted by max View Post
    Find theta and hence the angle of rotation and the direction of

    top row of matrix (-1/2) (1/2)
    bottom row (-1/2) (-1/2)

    Clockwise direction but
    when you calculate arccos (-1/2) you get 135 deg which is correct but
    arcsin (-1/2) is -45 deg

    What am I missing?

    Max

    matrix attached
    \cos \theta and \sin \theta are both negative in the third quadrant.

    So \theta = 225^0 or \theta = - 135^0.
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  7. #7
    max
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    Thank you for your reply and I understand where the -135 deg comes from. Its negative due to being a clockwise rotation about the origin (anti clockwise it is 225 deg as you say), where I am getting lost is why do I get -45 deg for calculating arcsin (-1/√2) ?
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    Quote Originally Posted by max View Post
    Thank you for your reply and I understand where the -135 deg comes from. Its negative due to being a clockwise rotation about the origin (anti clockwise it is 225 deg as you say), where I am getting lost is why do I get -45 deg for calculating arcsin (-1/√2) ?
    That's the correct answer, but to the wrong question!

    The correct question is:

    What value(s) of \theta simultaneously satisfy

    \cos \theta = -\frac{1}{\sqrt{2}} .... (1)

    \sin \theta = -\frac{1}{\sqrt{2}} .... (2)
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  9. #9
    max
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    Ahh okay thank you that actually makes it so much easier to understand.

    Max
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