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Rotating vectors and matrices

  1. Jul 9, 2008 #1
    I want to find the rotations needed to rotate one unit vector into another unit vector and then use these rotations to rotate a 3x3 matrix.

    For example: I want to determine the rotations needed to rotate [1 0 0] into [-0.342, -0.938, 0.0566] and apply the same rotation to the matrix [tex]M[/tex] =

    (1 0 0)
    (0 2 0)
    (0 0 3)

    The way I've thought of doing this is to:

    1. Rotate [1 0 0] about the z-axis by the angle arctan( [tex]\frac{0.938}{0.342}[/tex] ) to get [-0.3425 -0.9395 0]. Apply the same rotation to [tex]M[/tex].
    2. Take the cross product between [-0.3425 -0.9395 0] and [-0.342 -0.938 0.0566] to get a new axis of rotation [tex]\hat{r}[/tex].
    3. The new angle of rotation should be [tex]\hat{\theta}[/tex] = arctan([tex]\frac{0.0566}{\sqrt{0.3425^{2} + 0.9395^{2}}}[/tex]).
    4. Apply Rodriguez's rotation formula by [tex]\hat{\theta}[/tex] about [tex]\hat{r}[/tex] to [tex]M[/tex]

    I hope it's clear what I'm trying to do. If anyone can confirm that I'm doing this correctly, or come up with a better way of doing this, I'd very much appreciate it.

  2. jcsd
  3. Jul 12, 2008 #2
    Hello and welcome to PF!
    To rotate one vector into another one, you
    need only a single rotation in the plane containing the two vectors.
    Here is a general method of finding such a single rotation.

    The angle of rotation is a
    bivector whose direction specifies the plane of rotation and whose
    magnitude specifies how much to rotate.

    Let [tex] a [/tex] and [tex] b [/tex] be two unit vectors in 3D space.
    Since any unit vector multiplied by itself is just equal to the
    square of its magnitude, [tex] a^2=b^2=1 [/tex] , so it follows that
    a = ab^2 = (ab) b = (a \cdot b + a \wedge b) b.
    i.e. the geometric product [tex] ab [/tex] rotates the vector [tex] b [/tex]
    into the vector [tex] a [/tex] . The product may be written in terms of the
    angle of rotation, the bivector [tex] \mathbf A [/tex] .
    Write [tex] {\mathbf A} = \mid {\mathbf A} \mid \widehat{\mathbf A} [/tex] , where
    [tex] \mid {\mathbf A}\mid [/tex] is the magnitude of the
    rotation angle and [tex] \widehat{\mathbf A} [/tex] is the unit
    bivector specifying the plane of rotation. Using the fact that
    [tex] \widehat{\mathbf A}^2 = -1 [/tex] , the product can be expressed as
    ab = e^{\mathbf A}= \cos{\mathbf A} + \sin{\mathbf A}.
    ab = \cos{\theta} + \widehat{\mathbf A } \sin{\theta}.
    It only remains to identify the scalar and bivector parts of this with
    [tex] a\cdot b [/tex] and [tex] a\wedge b [/tex] (which you know) in order
    to get the sine and cosine of the rotation angle and the plane of

    Any vector in the rotation plane, [tex] \widehat{\mathbf A} [/tex],
    may therefore be rotated through the angle [tex] \theta [/tex] by
    pre-multiplying it with the geometric product [tex] ab [/tex] . Any
    vector perpendicular to this plane remains unaltered by this
    multiplication; hence, to rotate some arbitrary vector [tex] x [/tex] in the same
    way that you rotated the vector [tex] b [/tex] , you must first find
    its components parallel and perpendicular to the plane of rotation:
    [tex] x = x_\parallel + x_\perp
    The rotated vector is then
    [tex] x' = x_\perp + ab x_\parallel [/tex] .
    You can get the two components from
    [tex] x_\parallel = (x\cdot \widehat{\mathbf A})\widehat{\mathbf A}^{-1}
    [tex] x_\perp = (x\wedge \widehat{\mathbf A})\widehat{\mathbf A}^{-1}
    [/tex] .

    There is an alternative way to rotate an arbitrary vector. Let
    R = \cos{\theta/2} + \widehat{\mathbf A} \sin{\theta/2}
    R^\dagger = \cos{\theta/2} - \widehat{\mathbf A } \sin{\theta/2}
    The rotated vector is then
    [tex] x' = R^\dagger x R
    [/tex] .

    That's it, but it may be useful to spell this out somewhat. Let
    [tex] {e_1,e_2,e_3} [/tex] be a set of orthonormal vectors spanning the
    space. These have the properties [tex] e_i^2=1 [/tex]
    and [tex] e_i e_j = -e_j e_i [/tex] . The unit vectors defining the rotation
    are then
    a = a_1e_1 + a_2e_2 + a_3e_3
    b = b_1e_1 + b_2e_2 + b_3e_3
    The dot product and wedge products are
    a\cdot b = a_1b_1 + a_2b_2 + a_3b_3 = \cos{\theta}
    a\wedge b = {\mathbf A} = A_3 e_1e_2 + A_1 e_2e_3 + A_2 e_3e_1
    where [tex] A_3=a_1b_2 - a_2b_1 [/tex] , with similar expressions for
    [tex] A_1 [/tex] and [tex] A_2 [/tex] . The magnitude of the bivector
    [tex] \mathbf A [/tex] is
    [tex] \mid {\mathbf A} \mid = \sqrt{A_1^2+A_2^2 + A_3^2}=\sin{\theta} [/tex]
    and the unit plane of rotation is
    \widehat{\mathbf A}= {\mathbf A} / \sin{\theta}.
    This should be enough to get you started.
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