Rotation matrix Definition and 82 Threads

Homework Statement Find the eigenvalues and normalized eigenvectors of the rotation matrix cosθ -sinθ sinθ cosθ Homework Equations The Attempt at a Solution c is short for cosθ, s is short for sinθ I tried to solve the characteristic polynomial (c-λ)(c-λ)+s^2=0, and...

Homework Statement The components of stress in the x_i reference Cartesian system at a point of interested have been determined to be: \left[\begin{array}{ccc} 500 & 0 & 300 \\ 0 & 700 & 0 \\ 300 & 0 & -100 \end{array}\right] \mathrm{MPa} Determine the principal values and directions of...

Shankar 12.4.4 - "the" rotation matrix vs. "a" rotation matrix (tensor operators QM) Homework Statement My question comes up in the context of Shankar 12.4.4. See attached .pdf. Homework Equations See attached .pdf The Attempt at a Solution See attached .pdf I have this...

Can you calculate eigenvalues and eigenvectors for rotation matrices the same way you would for a regular matrix? If not, what has to be done differently?

Homework Statement I'm given an angular velocity tensor as shown. From this, I can extract the angular velocity. As I understand it, the angular velocity tensor matrix is the derivative of the rotation matrix. I have to determine the rotation matrix after t seconds. Homework Equations...

I know that a proper orthogonal rotation matrix in R^{2} has the form [cos \theta sin \theta -sin \theta cos \theta] which would rotate a vector by the angle \theta. However, I have also seen the matrix [sin \theta cos \theta -cos \theta sin \theta] What type of rotation...

Hello. Sorry for my English There are R - rotation matrix (that performs transformation from associated coordinate system IE to static coordinate system OI) and \omega - angular velocity. The matrix R depends on parameters \xi (for example, Euler angles). I need to express \omega as function...

I have a set of given vectors, I want to find a rotation matrix to convert them to vectors belong to surface of a cone with vertex is origin(vectors with the same slant angle but different tilt angles). Is there anybody know what is the solution? Thanks

Hi, I have a rotated frame (new matrix, T(x,y, z)) and the original frame (old matrix T(X,Y,Z)). I want to use this formula to find the Rotation matrix: T(x,y, z) = R-1 T(X,Y,Z) R Is this equation right? how can I calculate R (rotation matrix) in ZYZ order in this equation? Thank You

Homework Statement This question is from Principles of Quantum Mechanics by R. Shankar. Given the operator (matrix) \Omega with eigenvalues e^{i\theta} and e^{-i\theta} , I am told to find the corresponding eigenvectors.Homework Equations \Omega = \left[ \begin{array}{cc} \cos{\theta}...

Hello all, I am having a problem with this question. Can not see what I am doing wrong. Homework Statement Show that the two expressions are equivalent, by construction a rotation matrix Rsi. S = (-R sin(a*a_dot) - w R sin(a))s1 + (R cos(a*a_dot) + w R cos(a))s2 I = (-R...

I'm attempting to do some problems in a group theory exercise for the first time and am falling flat on my face. Here's the problem: "the molecule 'triangulum' consists of 3 identical atoms arranged in an equilateral triangle. Using a basis which consists of a single localised orbital on each...

Hi All, I have a rigidbody simulation and I'm trying to calculate the local angular velocity of the object using the derivative of it's orthogonal rotation matrix. This is where I'm stuck as I haven't been able to find an example on calculating the time derivative from two matrices at t=n and...

I feel I may have improperly posted this thread https://www.physicsforums.com/showthread.php?t=469331" but am just not as knowledgeable in my matrix math as I need to be. One (me) would think that somehow you should be able to get a rotation matrix from these two systems. Homework Statement So...

Dear friends, I've come across this questions when studying biatomic molecules. Here's my problem: You have the following two wave functions: Psi_1 = px(A) + px(B) Psi_2 = py(A) + py(B) here px(A) is the px orbital wave function of the A nucleus, px(B) of the B nucleus and so on...

Homework Statement Find the transformation matrix R that describes a rotation by 120 about an axis from the origin through the point (1,1,1). The rotation is clockwise as you look down the axis toward the origin. Homework Equations Rotations about the z-axis are given by R_{z}(\alpha) =...

I have a shape with spherical coordinate (r, theta, phi) which I can convert to Cartesian. I want to apply rotation to the shape by incrementing theta & phi. I figured out the matrix for rotating azimuth angle is { {cos(theta), -sin(theta), 0} {sin(theta), cos(theta), 0} { 0, 0, 1} } How...

Hello every one, I am a new comer. During my research of ionization rate of molecule using ADK method, I meet a question. What is the rotation matrix and R the Euler angles between the molecular axis (in Eq. (8) of reference PHYSICAL REVIEW A, 66, 033402 (2002)) and what form is the...

The following is my problem: I have a rotation and rotation matrix, based on rotations around coordinate A(x1,y1,z1). But actually, the rotation found place around coordinate B(x2,y2,z2). How can I adjust my rotation and translation matrix, so that it is adjusted for the rotations around...

It was pretty cool to stumble upon Euler's formula as the eigenvalues of the rotation matrix. det(Rot - kI) = (cos t - k)2 + sin2t =k2-2(cos t)k + cos2t + sin2t =k2-2(cos t)k + 1 k = {2cos t +/- \sqrt{4cos^2(t) - 4}}/2 k = cos t +/- \sqrt{cos^2(t) - 1} k = cos t +/- \sqrt{cos^2(t) - cos^2t -...

Homework Statement Let A be 2x2 and det(A)=1 and entries in R. Suppose A does not have any real eigenvalues. Then prove there exists a B st B is 2x2, det(B)=1 and BAB^-1=[cos(x),sin(x),-sin(x),cos(x)] for some x. The Attempt at a Solution I'm not sure how to start this proof. Any...

Homework Statement I need to find a 3x3 rotation matrix that takes a point in regular cartesian space and gives its coordinates in a rotate x`y`z` space. The +z` axis runs along the vector [1,1,-1], and the +x` axis should be in the xz plane with positive x component. Homework Equations The...

I am having some trouble deciphering what the input and output of a 2D Rotation Matrix actually represent. All example online have the vectors oriented at the origin. I know you can move them anywhere so long as you maintain their length and orientation, but here is my question: Let's say...

Say I have a matrix similar to the SO(3) matrix for general 3-D rotations, except it has slightly different (simpler) elements, and the symmetry is as follows: \left(\begin{array}{ccc} A & B & C \\ B & D & E \\ C & E & D \end{array}\right) , with A, B, C, D, and E all involving somewhat...

In Goldstein there is a problem asking to find a vector representation for a reflection in a plane of a unit normal \mathbf{\hat{n}}. I find it to be \mathbf{r'} = \mathbf{r} - 2(\mathbf{r\cdot \hat{n}})\mathbf{\hat{n}} and it has a corresponding transformation matrix with elements A_{ij} =...

I know how the rotation matrix looks like in the 2x2 and 3x3 orders, but how does it look in general? thanks in advance.

[SOLVED] Representation of j=1 rotation matrix The derivation of this involves the use of the following fact for j=1: [atex]\frac{J_y}{\hbar} = (J_y/\hbar)^3[/itex]. Is there a simple way to see this other than slogging through the algebra by expanding out the RHS using J_y =...

cos a -sin a sin a cos a How do I find the eigenvalue of this rotation matrix? I did the usual way, but didn't work! Could someone tell me how to start this problem?

I am having a hard time figuring this out. Suppose we have a 4x4 matrix A, B and rotation matrix D. Matrix A represent position and orinetation of object1, matrix B represent position and orientation of object2. Matrix D is the position and oreintation of object2 relative to object1. B = D*A...

Homework Statement A vector x in R^2 is rotate twice through an angle theta (it is rotated through theta and again through theta). Find two expressions for the matrix representing this rotation. Verify that these two expressions are equivalent Homework Equations rotation matrix R=[cos...

Problem 1.9 of DJGriffiths asks for the rotation matrix about the (1,1,1) direction. I thought I could rotate about z 45 degrees (R': x -> x'), then rotate about y' (R'': x' -> x''). How do I combine the two rotations to determine the final single rotation matrix... R = R''*R' or R = R'*R'' ...

Hi, if i want to find a 3x3 matrix R which represents a rotation of Pi/6 around the axis of rotation v(vector)={1, 2, 3}. how can i find it?