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Passive and Active Transformations

  1. Jun 12, 2015 #1
    Alright. I was looking into a 2D rotation matrix and there are two equations: one is through the transformation of the component of p (always with respect to x,y), x,y into x',y' and the other is through the transformation of the unit vectors i,j into i',j'.

    In a sense 1 is passive the other is an active transformation. I do understand the difference between the two, but my question is that why do we get the different matrices if when we derive them, we do so with a clockwise orientation?

    Hope its clear, If not--I'd be happy to clarify.
  2. jcsd
  3. Jun 12, 2015 #2
    Can you clarify. I guess I understand what you mean, but just to be sure.
  4. Jun 12, 2015 #3
    LOL Okay. I got the answer. Welp it seems like they way you derive the matrix you have to say consistent, otherwise you fall in a a trap.
    Staring at this image for a bit and making sense of this reasoning: http://math.stackexchange.com/quest...erent-representations-of-2d-rotation-matrices

    helped. You need to use a point relative to the rotated basis once you rotate the basis. You cannot use a rotated basis on a point relative to a the old basis, but with rotational matrices you only an inverted answer and nothing any more alarming.
  5. Jun 12, 2015 #4


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    See this thread for some general comments about "active" vs. "passive" rotations.

    The convention that I'm familiar with is to rotate vectors counterclockwise. A rotation is linear, so if we know how the basis vectors are rotated, we will know how all vectors are rotated. If you rotate (1,0) by an angle ##\theta## counterclockwise, the result is ##(\cos\theta,\sin\theta)##. The same rotation applied to (0,1) yields ##(-\sin\theta,\cos\theta)##. It follows immediately from this that the matrix of the rotation operator with respect to the standard ordered basis is
    $$\begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{pmatrix}.$$ The rotated basis vectors end up as the columns of the matrix. If you don't understand this, I recommend the FAQ post that I linked to in the post I linked to above.
    Last edited: Jun 12, 2015
  6. Jun 12, 2015 #5
    Hehe alright, thanks. Looking back at it, I was being pretty dense.
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