1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Linear Transformation: find dilating/rotation matrix

  1. Sep 13, 2012 #1
    1. The problem statement, all variables and given/known data

    The vector A has length 8.5, and makes an angle of 5pi/19 with the x-axis.

    The vector B has length 6, and makes an angle of 8pi/19 with the x-axis.

    Find the matrix which rotates and dilates vector into vector .

    2. Relevant equations
    Rotation matrix in counterclockwise direction: \begin{equation}

    \left[
    \begin{array}{ccc}
    cos∅ & -sin∅\\
    sin∅ & cos∅\\
    \end{array}
    \right]

    \end{equation}

    Dilation matrix:
    \begin{equation}

    \left[
    \begin{array}{ccc}
    c & 0\\
    0 & c\\
    \end{array}
    \right]

    \end{equation}


    3. The attempt at a solution

    \begin{equation}

    \left[
    \begin{array}{ccc}
    (12/17)cos(3pi/19) & -sin(3pi/19)\\
    sin(3pi/19) & (12/17)cos(3pi/19)\\

    \end{array}
    \right]

    \end{equation}

    My line of thinking: to move A into B, it needs to move counterclockwise from its current positions by an angle of 3pi/19. A needs to "shrink" by a factor of 12/17.

    However, this is incorrect. I would really appreciate it if someone could point me in the right direction, or even give me a similar example. Thank you.
     
  2. jcsd
  3. Sep 13, 2012 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    You can either rotate first, then shrink, or shrink first, then rotate. Also, be careful about rotations: for rotation a coordinate system through angle θ but leaving a vector v = <x,y> unchanged, the coordinates x' and y' of v in the new coordinate system are given by
    [tex] \pmatrix{x' \\ y'} = \pmatrix{\cos(\theta) & -\sin(\theta)\\
    \sin(\theta) & \cos(\theta)} \pmatrix{x\\y}.[/tex]
    However, if we rotate the vector v through angle θ but keeping the coordinate system unchanged, the coordinates x' and y' of the new vector v' are given by
    [tex] \pmatrix{x' \\ y'} = \pmatrix{\cos(\theta) & \sin(\theta)\\
    -\sin(\theta) & \cos(\theta)} \pmatrix{x\\y}.[/tex]
    Note that the two transformation matrices are transposes of each other.

    RGV
     
  4. Sep 13, 2012 #3

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    The way you combined the rotation and dilation isn't correct. As Ray said, do one first and then the other.
     
  5. Sep 13, 2012 #4
    To dilate:\begin{equation}

    \left[
    \begin{array}{ccc}
    (12/17) & 0\\
    0 & (12/17) \\

    \end{array}
    \right]

    \end{equation}

    To rotate:
    \begin{equation}

    \left[
    \begin{array}{ccc}
    cos(3pi/19) & -sin(3pi/19)\\
    sin(3pi/19) & cos(3pi/19)\\

    \end{array}
    \right]

    \end{equation}


    Does the above look right?

    So I'm trying to find matrix, a 2X2 matrix so that: \begin{equation}

    \left[
    \begin{array}{ccc}
    A\\
    \\

    \end{array}
    \right]

    *

    \left[
    \begin{array}{ccc}
    8.5cos(5pi/19)\\
    8.5sin(5pi/19)\\

    \end{array}
    \right]

    =

    \left[
    \begin{array}{ccc}
    6cos(8pi/19)\\
    6sin(8pi/19)\\

    \end{array}
    \right]

    \end{equation}

    Is there a way to put these two transformations together, into one matrix, A? Thank you for the help you have given so far.
     
  6. Sep 13, 2012 #5

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    In your original post, if you multiply your dilation matrix by your rotation matrix correctly, you should have a 12/17 times all four entries. Then try that for your A.
     
  7. Sep 13, 2012 #6
    Thank you LCKurts, vela, and Ray! So to summarize: think of the transformations separately, and then multiply the two transformation matrices together.
    Thanks again
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Linear Transformation: find dilating/rotation matrix
Loading...