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 x-axis). 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.
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.
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.