Rotation matrices Definition and 21 Threads

First of all I have this system $$\begin{pmatrix}\tau_{xx} \\ \tau_{yy} \\ \tau_{zz} \\ \tau_{xy} \\ \tau_{yz} \\ \tau_{zx} \end{pmatrix}=\begin{pmatrix}C_{11} & C_{12} & C_{12} & 0 & 0 & 0 \\ C_{12} & C_{11} & C_{12} & 0 & 0 & 0 \\ C_{12} & C_{12} & C_{11} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0...

The easiest proof I know for the 2D statement in the summary does not carry over nicely to the 3D statement since rotations in 3D don't necessarily commute (the 2D proof uses this commuting among rotations in the plane around a common point). Before I then try to modify the proof so that it...

I found a the answer in a script from a couple years ago. It says the kinetic energy is $$ T = \frac{1}{2} m (\dot{\vec{x}}^\prime)^2 = \frac{1}{2} m \left[ \dot{\vec{x}} + \vec{\omega} \times (\vec{a} + \vec{x}) \right]^2 $$ However, it doesn't show the rotation matrix ##R##. This would imply...

Does the position of the origin for the body’s rotating coordinate frame 1) stay fixed to the moving body or 2) does it stay fixed to the inertial frame, yet still able to rotate as the body rotates with the only restriction that it cannot translate with the body i.e. only affixed at the...

Hello Before I "phrase" my question (and that may be my problem), may I first state what I do know. I understand that a Rotation matrix (a member of SO(3)) has nine elements. I also understand that orthogonality imposes constraints, leaving only three free parameters (a sub-manifold) I also...

Homework Statement so I was going over my notes on classical mechanics and just started to review rotation matrices which is the first topic the book starts with. On page 3, I've uploaded the page here The rotation matrix associated with 1.2a and 1.2b is \begin{pmatrix} \cos\theta &...

Can anyone give me geometric and intuitive insight on Rotation matrices which has two sets of coordinates after Transformation?

Say I have {S_{x}=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc} 0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0\\ \end{array}\right)} Right now, this spin operator is in the Cartesian basis. I want to transform it into the spherical basis. Since, {\vec{S}} acts like a vector I think that I only need to...

First, I'd like to say I apologize if my formatting is off! I am trying to figure out how to do all of this on here, so please bear with me! So I was watching this video on spherical coordinates via a rotation matrix: and in the end, he gets: x = \rho * sin(\theta) * sin(\phi) y = \rho*...

I'm trying to rotate a point about the origin (0,0,0) and starting with an identity matrix, this works fine for the x- and y-rotation axes, but fails with the z-axis, where the point just sits in place. \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} M_{ID} \times M_Z...

Show that a rotation by θ followed by a rotation by φ can be expressed as either two consecutive rotations, or one rotation of (θ + φ). That is, show that Qθ Qφ = Qθ+φ, where Q is the rotation matrix. Can anyone answer this question I'm a beginner in Linear Algebra

I probably can remember the matrices by just trying to, but I hate having to "remember" things without actually understanding them. Is there no intuition behind these matrices so that I can remember it (the intuition) and then from it produce the wanted matrix? To me the matrices look like...

I also posted this in the homework help for introductory physics, but it wasn't getting any responses, so I guess it's slightly more advanced. Homework Statement Let L_b(a) denote the 4x4 matrix that gives a pure boost in the direction that makes an angle a with the x-axis in the xy plane...

Dear All, I have inherited a few rotation matrices through some old computer code I am updating. The code is used to construct some geometry. The matrices I have inherited are left handed rotation matrices and they are being applied to a right handed coordinate system, but they give the...

[URGENT] Rotation Matrices Homework Statement http://e.imagehost.org/0661/Screen_shot_2010-03-09_at_12_37_44_AM.png Homework Equations Rotation Matrix: cos(theta) -sin(theta) sin(theta) cos(theta) The Attempt at a Solution I understand 2a: cos(pi/4) -sin(pi/4) sin(pi/4)...

I am having a hard time figuring out 3d rotation matrices expression. After much search, I got 2D - http://www.siggraph.org/education/materials/HyperGraph/modeling/mod_tran/2drota.htm With 2d, x'=xcos t - ysin t y'=xsin t + ycos t and the matrix is : [cos t -sin t] [sin t cos t]...

I am looking for two rotation matrices M1 and M2, which describe a rotation by an arbitrary angle around the axes passing through (0,0,0) and (1,1,1), and (1,0,0) and (2,1,1). All relative to the standard basis. How would I approach this problem?

So, I've been fiddling around with a computer game (not the most productive use of my time, I know) and I've come across a problem that seems to have broader mathematical import, and most certainly has been found before, so I thought I'd ask about it here. Basically I need to rotate an object...

A long literature search has given me nothing, so I'm turning to this forum for help. I have a spin-3/2 fermion field, and I want to find its wave functions corresponding to its 4 pure-spin states, +3/2, +1/2, -1/2, -3/2, which is normally done by finding the 4 eigenfunctions of its rotation...

Hi, After obtaining the 2D rotation matrix (as a function of rotation angle) once by geometry and once by complex algebra, I tried to obtain it by invariance of the Euclidean metric. By this approach, the four elements of the 2D rotation matrix can be determined in terms of a single...

I'm working on a skeletal animation system, and I want to interpolate the rotations between frames. The rotations are represented by 3x3 matrices. It's easy enough to interpolate one of the axes linearly by averaging the values from the two frames and re-normalizing, but when you do this...