# What is Rotation matrices: Definition and 24 Discussions

In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix

R
=

[

cos

θ

sin

θ

sin

θ

cos

θ

]

{\displaystyle R={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}}
rotates points in the xy-plane counterclockwise through an angle θ with respect to the x axis about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates v = (x, y), it should be written as a column vector, and multiplied by the matrix R:

R

v

=

[

cos

θ

sin

θ

sin

θ

cos

θ

]

[

x

y

]

=

[

x
cos

θ

y
sin

θ

x
sin

θ
+
y
cos

θ

]

.

{\displaystyle R\mathbf {v} \ =\ {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}\ =\ {\begin{bmatrix}x\cos \theta -y\sin \theta \\x\sin \theta +y\cos \theta \end{bmatrix}}.}
If x and y are the endpoint coordinates of a vector, where x is cosine and y is sine, then the above equations become the trigonometric summation angle formulae. Indeed, a rotation matrix can be seen as the trigonometric summation angle formulae in matrix form. One way to understand this is say we have a vector at an angle 30° from the x axis, and we wish to rotate that angle by a further 45°. We simply need to compute the vector endpoint coordinates at 75°.
The examples in this article apply to active rotations of vectors counterclockwise in a right-handed coordinate system (y counterclockwise from x) by pre-multiplication (R on the left). If any one of these is changed (such as rotating axes instead of vectors, a passive transformation), then the inverse of the example matrix should be used, which coincides with its transpose.
Since matrix multiplication has no effect on the zero vector (the coordinates of the origin), rotation matrices describe rotations about the origin. Rotation matrices provide an algebraic description of such rotations, and are used extensively for computations in geometry, physics, and computer graphics. In some literature, the term rotation is generalized to include improper rotations, characterized by orthogonal matrices with a determinant of −1 (instead of +1). These combine proper rotations with reflections (which invert orientation). In other cases, where reflections are not being considered, the label proper may be dropped. The latter convention is followed in this article.
Rotation matrices are square matrices, with real entries. More specifically, they can be characterized as orthogonal matrices with determinant 1; that is, a square matrix R is a rotation matrix if and only if RT = R−1 and det R = 1. The set of all orthogonal matrices of size n with determinant +1 forms a group known as the special orthogonal group SO(n), one example of which is the rotation group SO(3). The set of all orthogonal matrices of size n with determinant +1 or −1 forms the (general) orthogonal group O(n).

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1. ### I Also in 3D, two reflections make a rotation?

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...
2. ### Square of orthogonal matrix vanishes

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...
3. ### I Relative Motion & Local Frame’s Position - when projecting components

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...
4. ### A How do I KNOW that Euler angles are sufficient?

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...
5. ### Possible error in Marion and Thornton's Classical Dynamics?

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 &...
6. L

### B Exploring Rotation Matrices: Intuitive Insights and Geometric Interpretation

Can anyone give me geometric and intuitive insight on Rotation matrices which has two sets of coordinates after Transformation?
7. ### Generators of Lie Groups and Angular Velocity

I envision the three fundamental rotation matrices: R (where I use R for Ryz, Rzx, Rxy) I note that if I take (dR/dt * R-transpose) I get a skew-symmetric angular velocity matrix. (I understand how I obtain this equation... that is not the issue.) Now I am making the leap to learning about...
8. ### A Transforming Spin Matrices (Sx, Sy, Sz) to a Spherical Basis

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...
9. ### I Spherical coordinates via a rotation matrix

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*...
10. ### 3x3 Identity and rotation matrices and how they work

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...
11. ### Linear Algebra: Rotation Matrix Qθ+φ

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
12. ### Intuition behind rotation matrices?

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...
13. ### Lorentz boosts and rotation matrices

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...
14. ### Lorentz boosts and rotation matrices

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. Explain why this can be found as L_b(a) = L_r(-a)*L_b(0)*L_r(a), where L_r(a) denotes the matrix that rotates the xy plane through the angle a and...
15. ### Converting Rotation Matrices (Left handed to Right handed)

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...
16. ### Solving Rotation Matrices Urgently: cos(pi/4) -sin(pi/4) sin(pi/4) cos(pi/4)

[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)...
17. ### Derivation of non-inertial terms in non-inertial systems using rotation matrices

(I know how to do this without the rotation matrices) Any suggestion would be much appreciated. Homework Statement Show that the relationship between the forces in the inertial (S') and non-inertial(S) reference frames, with a coordinate transformation given by \vec{r}=R \vec{r'}...
18. ### Solving 3D Rotation Matrices: A Step-by-Step Guide

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]...
19. ### Website title: Finding Rotation Matrices for Arbitrary Angle Rotations in R^3

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?
20. ### Solve Rotation Matrices w/ X, Y, Z Axes

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...
21. ### What is the Operator for Spin-3/2 Rotation in Fermion Fields?

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...
22. ### Determinant of rotation matrices

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...
23. ### Exploring Rotation Matrices: Finite & Infinitesimal Rotations

Homework Statement Can anyone help me to proceed with this? If we execute rotations of 90* about x-axis and 90* about y axis-what is the resulatant rotation matrix?Will the result commute if we rotate by changing the order?Will they commute if infinitesimal rotations are considered...
24. ### Interpolating Rotation Matrices

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...