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    Fundamental Mathematics: Rotation & Reflection

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    We use matrices, eigenvalues and eigenvectors to solve the following:

    Let f be the rotation in the 3-dimensional space about the x-axis through the angle pi/2 and g be the
    rotation about the y-axis through the same angle. Describe the rotation f g. Let h be the reflection in the plane orthogonal
    to the vector 2i - j + 3k. Describe the reflection f h.

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    Exercise: Let f be the rotation in the 3-dimensional space about the x-axis through the angle pi/2 and g be the
    rotation about the y-axis through the same angle. Describe the rotation f g. Let h be the reflection in the plane orthogonal
    to the vector 2i - j + 3k. Describe the reflection f h.

    It is not specified whether the rotation f is clockwise or counterclockwise, so we assume counterclockwise, that is, it takes
    the vector (0,1,0)^T to the vector (0,0,1)^T (here v^T means transpose). The matrix associated to f is then

    A=...

    since alpha=frac{pi}{2}. Similarly for g, we take the rotation by frac{pi}{2} about the y-axis that takes the
    vector (0,0,1)^T to (1,0,0)^T. It has matrix

    B=...

    Note: if g rotates in the opposite direction them multiply the above matrix by -1 (or equivalently take alpha=-frac{pi}{2}
    ).

    The matrix for fg is the product C=AB

    C=...

    This is again a rotation matrix. To find the axis of rotation find v such that Cv=v (so v is fixed under C, meaning it is
    the axis of rotation). Cv=v is equivalent to (C-I)v=0.

    C-I=...

    So (C-I) multiplied by the vector (x,y,z)^T gives
    begin{align*}
    -x+z=0,;x-y=0,;y-z=0
    end{align*}
    that is x=z,y=z (and z=z). We see that the vector v=(1,1,1)^T makes (C-I)v=0. Then fg is rotation about (1,1,1)^T. We
    need to know the angle of rotation. Since C fixes the space orthogonal to v we take a basis for that space to find the angle
    of rotation.

    Note that v_1=frac{1}{sqrt{2}}(1,0,-1)^T, v_2=frac{1}{sqrt{6}}(-1,2,-1) and v_3=frac{1}{sqrt{3}}(1,1,1) are orthogonal
    vectors and of unit length. Then frac{1}{sqrt{2}}(1,0,-1)^T and frac{1}{sqrt{6}}(-1,2,-1) form an orthonormal basis for
    the orthogonal space to frac{1}{sqrt{3}}(1,1,1) that is fixed by C. To find the angle of rotation there are different ways.
    One is to find the representation of C in this new basis.

    First note that
    Cv_1=...

    Cv_2=...

    Cv_3=v_3

    Then

    C(av_1+bv_2+cv_3)=...

    so (a,b,c)^T is sent to (-frac{a}{2}-frac{sqrt{3}b}{2},frac{sqrt{3}a}{2}-frac{b}{2},c). The matrix involved is

    ...

    which is of the form

    ...

    for alpha=frac{2pi}{3}.

    Therefore fg is rotation about the axis (1,1,1)^T by angle frac{2pi}{3}. (You can figure out which way it acts, if
    counterclockwise or clockwise by calculating Cv_1 or Cv_2 and see which way it turns, I leave it to you).

    For the reflection fh. The method is similar. First find a matrix representation of h. One way to do that is to note that if
    we call D the matrix of h, then Du_1=-u_1 for u_1=2i - j + 3k=(2,-1,3)^T. Also h fixes the plane orthogonal to u_1.
    The vectors u_2=(1,2,0)^T and u_3=(0,3,1)^T are orthogonal to u_1 so Du_2=u_2 and Du_3=u_3. We want to know what is
    D(x,y,z)^T. We know
    [Du_1=-u_1,,Du_2=u_2,,Du_3=u_3,]
    or equivalently
    [D[u_1,u_2,u_3]=[-u_1,u_2,u_3],]
    where [u_1,u_2,u_3] is the matrix whose columns are u_1,u_2,u_3. Call E=[u_1,u_2,u_3]. Then D=[-u_1,u_2,u_3]E^{-1}

    D=...

    To identify the reflection D, find u such that Du=-u, that is (D+I)u=0. Solving as before we see that w_1=(2/3, -1/3,
    1)^T. So fh is reflection with respect to the plane orthogonal to (2/3, -1/3,1)^T.

    Note that D fixes the vectors w_2=(-3/2, 0, 1)^T and w_3=(1/2, 1, 0) and that w_2 and w_3 are orthogonal to w_1 (so
    we see that D does fix the space orthogonal to w_1, and so it is a reflection).

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 10:53 pm ad1c9bdddf>
    https://brainmass.com/math/linear-algebra/fundamental-mathematics-rotation-reflection-516950

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