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[Linear Algebra] rotational matrices

  1. Jul 24, 2016 #1
    1. The problem statement, all variables and given/known data
    Prove Rθ+φ =Rθ+Rφ Where Rθ is equal to the 2x2 rotational matrix
    [cos(θ) sin(θ),
    -sin(θ) cos(θ)]

    2. Relevant equations
    I am having a hard time trying figure our what is being asked. My question is can anyone put this into words? I am having trouble understanding what the phi may mean?

    3. The attempt at a solution
    I tried to solve this graphical by plotting x and y vectors doing a rotation and plotting those and taking the product of the two rotations or x and y and finding the rotation of x+y. I don't think this is even on the right track and there must be a much simpler way to do this than graphically.
     
  2. jcsd
  3. Jul 24, 2016 #2

    fresh_42

    Staff: Mentor

    A proper graphic can at least show you the way how to do it. You need only one vector at the center of rotation and then rotate it in two steps, e.g. by using a compass. Do you know what the consecutive application of two rotations mean to the two rotation matrices?
     
  4. Jul 24, 2016 #3

    micromass

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    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    This seems wrong. Are you sure that you copied it correctly?
     
  5. Jul 24, 2016 #4
    You are write that it isn't correct it should be Rθ+φ =Rθ*Rφ
    THANK YOU!!!
     
  6. Jul 24, 2016 #5

    fresh_42

    Staff: Mentor

    And have you multiplied the two?
     
  7. Jul 24, 2016 #6
    No I have not because I am unsure as to what Rφ is.
     
  8. Jul 24, 2016 #7
    No I am not familiar with what two applications of the rotation means.
     
  9. Jul 24, 2016 #8

    fresh_42

    Staff: Mentor

    The same as with ##θ##. So ##R_φ = \begin{bmatrix}\cos φ && \sin φ \\ -\sin φ && \cos φ\end{bmatrix}##.
    It means to do one rotation by ##φ## applied to some original vector ##v## and then another rotation by ##θ## applied to the resulting vector of the first.
    In terms of matrices, it is - as you wrote - ## (R_θ \cdot R_φ) (v) = R_θ ( R_φ (v)) ##, i.e. you will have to multiply the two matrices.
    The addition theorems for ##\cos## and ##\sin## should give you the required result.

    And you can (and in my opinion should) do all this graphically, too. It will give you a nice proof of the addition theorems.
     
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