# rotation matrix

by dirk_mec1
Tags: matrix, rotation
 P: 664 1. The problem statement, all variables and given/known data The rotation matrix below describes a beam element which is rotated around three axes x,y and z. Derive the rotation matrix. 2. Relevant equations - 3. The attempt at a solution I can see where the x-values (CXx CYx CZx) come from. They're just the projections of the rotated x-axes (the one with rotation alpha and beta). But I don't understand how the rest is derived can somebody help me?
 Math Emeritus Sci Advisor Thanks PF Gold P: 38,898 Rotation about the x-axis through angle $\alpha$ is given by the matrix $$\begin{bmatrix}1 & 0 & 0 \\ 0 & cos(\alpha) & -sin(\alpha) \\ 0 & sin(\alpha) & cos(\alpha)\end{bmatrix}$$ Rotation about the y-axis through angle $\beta$ is given by the matrix $$\begin{bmatrix}cos(\beta) & 0 & -sin(\beta) \\ 0 & 1 & 0 \\ sin(\beta) & 0 & cos(\beta)\end{bmatrix}$$ Rotation about the z-axis through angle $\gamma$ is given by the matrix $$\begin{bmatrix} cos(\gamma) & -sin(\gamma) & 0 \\ sin(\gamma) & cos(\gamma) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ The result of all those rotations is the product of those matrices. Be sure to multiply in the correct order.
 P: 664 I suspect that there's a minus sign somewhere wrongly placed in your matrices Halls, am I correct? I moved the minus sign in your second matrix to the lower sine but there's still something wrong for this is my result: [ cos(a)cos(b), -sin(b), cos(b)sin(a) ] [ sin(a)sin(c) + cos(a)cos(c)sin(b) cos(b)cos(c) cos(c)*sin(a)sin(b) - cos(a)sin(c) ] [ cos(a)sin(b)sin(c) - cos(c)sin(a) cos(b)*sin(c) cos(a)cos(c) + sin(a)sin(b)sin(c) ]
Math
Emeritus
Thanks
PF Gold
P: 38,898

## rotation matrix

No, all of the minus signs are correctly placed. I am, of course, assuming that a positive angle gives a rotation "counterclockwise" looking at the plane from "above"- from the positive axis of rotation.
 P: 664 But the wiki page shows a different position for the minus sign of your second matrix: http://en.wikipedia.org/wiki/Rotation_matrix.
Mentor
P: 14,476
 Quote by dirk_mec1 1. The problem statement, all variables and given/known data The rotation matrix below describes a beam element which is rotated around three axes x,y and z. Derive the rotation matrix.
Look at your diagram. Are all of those rotations positive by the right hand thumb rule? (Hint: The answer is no.)

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