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Transformation Matrix Problem 
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#1
Nov305, 08:59 AM

P: 31

I don't know if this is the right section, but this problem is in my electromagnetism course (Griffiths text).
This is problem 1.9 of Griffiths (3rd edition) text: Find the transformation matrix R that describes a rotation by 120 degrees about an axis from the origin through the point (1,1,1). The rotation is clockwise as you look down the axis toward origin. At first, I didn't understand the question (actually I still think I don't understand it). But then I read a book on vectors and tensors that I used as a reference in my vector analysis course last year. I couldn't come up with a solution even then. So I discussed it with a friend. We came up with the following solution: (phi=120degrees) (cos phi sin phi) = (0.5 0.866) (sin phi cos phi) (0.866 0.5) I am sorry I can't present it in LaTeX as I am not experienced, but there are square matrices on boths sides of the equation. But isn't this ridiculously simple? This just tells of rotation of 120 degrees about a certain axis (say xaxis). It doesn't say anything about a new axis coming from origin to point (1,1,1). What more should I do? A friend put the two components Ay, Az both equal to 1. Multiplied this column vector with the rotation matrix I wrote above, and came up with the value of Ay(prime) and Az(prime). His values were: Ay(prime)=0.366 Az(prime)=1.366 But this gives the value of coordinates, not the rotation matrix itself. The question asks for the rotation matrix! I simply don't understand what I should do with this question. This is basically a chapter on vector analysis, and I have almost done all other questions except this one. This one has been bothering me for days now. I have no idea if my solution is right. I don't even know why the question specifically mentions an axis through (1,1,1), if it only required me to put rotation angle, phi, equal to 120degree! Any help will be greatly appreciated. 


#2
Nov305, 12:32 PM

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P: 2,002

The idea is that, since we use vectors to denote physical quantities and we want to manipulate the vectors with components relative to a coordinate system, it shouldn't matter how we choose such a coordinate system. And we want to switch (transform) a set of components wrt one coordinate system to another. That's what the transformation matrix does.
Now suppose you have some vector [itex]\vec A[/itex] and you have two coordinate systems [itex]O[/itex] and [itex]O'[/itex]. The origins coincide, but the axes of [itex]O'[/itex] are rotated with respect to [itex]O[/itex] (it's rotated 120 degrees about the axis in the (1,1,1) direction in [itex]O[/itex]). The transformation matrix should tell you how to transform the components of the vector [itex]\vec A[/itex] in [itex]O[/itex], which are [itex](A_x,A_y,A_z)[/itex] into the components wrt O': [itex](A_x',A_y',A_z')[/itex]. I hope that'll help you in understanding the question. You can already smell immediately that you'll need 3x3 matrix. 


#3
Sep108, 04:24 PM

P: 15

Greetings
I am in this course now and have this same problem. I know the rotation matrix is supposed to be a 3x3 one and the formula in the book shows: Rxx Rxy Rxz Ryz Ryy Ryz Rzx Rzy Rzz I don't understand what to do next, what does Rxx and Rxy mean? I was wondering if someone could give an example of one of these variables just so I know what it is supposed to look like. The book doesn't really explain much about this. 


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