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I dont get how these matrices are mulitiplied together

  1. Feb 5, 2012 #1
    1. The problem statement, all variables and given/known data
    H10 =
    c1 | -s1 | 0
    s1 | c1 | 0
    0 | 0 | 1

    H21 =
    c2 | -s2 | 0
    s2 | c2 | 0
    0 | 0 | 1



    2. Relevant equations
    this is what i did

    H10H21 =
    c1c2 -s1s2 | -c1s2 - s1c2 | 0
    s1c2 + c1s2 | -s1s2 + c1c2 | 0
    0 | 0 | 1


    3. The attempt at a solution

    this is the answer

    H10H21 =

    c1c2 | -s1s2 | 0
    s1s2 | c1c2 | 0
    0 | 0 | 1
     
    Last edited: Feb 5, 2012
  2. jcsd
  3. Feb 5, 2012 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Unless that is some other information that you have not given, your answer is correct.
    Note, by the way, that if [itex]c_1^2+ s_1^2= 1[/itex] and [itex]c_2^2+ s_2^2= 1[/itex], or if you divide each term of the matrices by that, we can interpret [itex]c_1[/itex] and [itex]s_1[/itex] as the cosine and sine of some angle, [itex]\theta[/itex] and can interpret [itex]c_2[/itex] and [itex]s_2[/itex] as the cosine and sine of some other angle, [itex]\phi[/itex] and so these two matrices as rotation, about the z-axis, through those angles. The product would be the combination of those two rotations, a rotation through angle [itex]\theta+ \phi[/itex] and then we have [itex]c_1c_2- s_1s_2= cos(\theta)cos(\phi)- sin(\theta)sin(\phi)= cos(\theta+ \phi)[/itex] and [itex]s_1c_1+ c_1s_2= sin(\theta)cos(\phi)+ cos(\theta)sin(\phi)= sin(\theta+ \phi)[/itex] as we should have.

    (What you give as the "answer" is sometimes called the "component by component" product but it does NOT have any good algebraic properties and is very seldom used.)
     
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