What is Rotation matrix: Definition and 81 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












{\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:





























{\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. BerriesAndCream

    I Can I visualize O(3) \ SO(3) in some way?

    Hello. I know that a 3×3 orthogonal matrix with determinant = 1 (so a 3×3 special orthogonal matrix) is a rotation in 3D. I was wondering if there is a 3×3 orthogonal matrix with determinant = –1 could be visualised in some way. Thank you!
  2. D

    Rotation matrix of three intrinsic rotations

    I have three frames. The first is the fixed global frame. the second rotates an angle PHIZ with respect to the first. And the third first rotates a PHIX angle with respect to the x axis of the second frame, and then rotates a PHIY angle with respect to the last y axis. That is, there are a total...
  3. K

    A Ambiguity in sense of rotation given a rotation matrix A

    Goldstein 3rd Ed pg 161. Im not able to understand this paragraph about the ambiguity in the sense of rotation axis given the rotation matrix A, and how we ameliorate it. Any help please. "The prescriptions for the direction of the rotation axis and for the rotation angle are not unambiguous...
  4. K

    A Rotation matrix and rotation of coordinate system

    If we change the orientation of a coordinate system as shown above, (the standard eluer angles , ##x_1y_1z_1## the initial configuration and ##x_by _b z_b## the final one), then the formula for the coordinates of a vector in the new system is given by ##x'=Ax## where...
  5. dontknow

    Constraints in Rotation Matrix

    In Rigid body rotation, we need only 3 parameters to make a body rotate in any orientation. So to define a rotation matrix in 3d space we only need 3 parameters and we must have 6 constraint equation (6+3=9 no of elements in rotation matrix) My doubt is if orthogonality conditions...
  6. brotherbobby

    On the orthogonality of the rotation matrix

    Let me start with the rotated vector components : ##x'_i = R_{ij} x_j##. The length of the rotated vector squared : ##x'_i x'_i = R_{ij} x_j R_{ik} x_k##. For this (squared) length to be invariant, we must have ##R_{ij} x_j R_{ik} x_k = R_{ij} R_{ik} x_j x_k = x_l x_l##. If the rotation matrix...
  7. Athenian

    Finding the Rotation Matrix for Vector Alignment

    Below is the attempted solution of a tutor. However, I do question his solution method. Therefore, I would sincerely appreciate it if anyone could tell me what is going on with the below solution. First off, the rotation of the matrix could be expressed as below: $$G = \begin{pmatrix} AB & -||A...
  8. T

    A Algebraic proof that Euler angles define a proper rotation matrix

    I have asked this question twice and each time, while the answers are OK, I am left dissatisfied. However, now I can state my question properly (due to the last few responses). Go to this page and scroll down to the matrix for sixth row of the proper Euler angles...
  9. H

    B What Does Skew Symmetry Imply for One-Dimensional Systems?

    Hi, if I have a equation like (just a random eq.) p_dot = S(omega)*p. where p = [x, y, z] is the original states, omega = [p, q, r] and S - skew symmetric. How does the equation appear if i only want a system to have the state z? do I get z_dot = -q*x + p*y. Or is the symmetric not valid so I...
  10. T

    Non-Rotation Matrix Split: Hello

    Hello This could very well be an idiotic question, but here goes... Consider a general matrix M Consider a rotation matrix R (member of SO(2) or SO(3)) Is it possible to split M into the product of a rotation matrix R and "something else," say, S? Such that: M = RS or the sum M = R + S...
  11. JTC

    A Example of how a rotation matrix preserves symmetry of PDE

    Good Day I have been having a hellish time connection Lie Algebra, Lie Groups, Differential Geometry, etc. But I am making a lot of progress. There is, however, one issue that continues to elude me. I often read how Lie developed Lie Groups to study symmetries of PDE's May I ask if someone...
  12. M

    Prove all Elements of O(2,R) have form of Rotation Matrix

    Homework Statement Show that every matrix A ∈ O(2, R) is of the form R(α) = cos α − sin α sin α cos α (this is the 2d rotation matrix -- I can't make it in matrix format) or JR(α). Interpret the maps x → R(α)x and x → JR(α)x for x ∈ R 2 Homework EquationsThe Attempt at a Solution So I know...
  13. S

    I Rotation Matrix for Vector v=(a,b,c) by Angle θ | Efficient Computation Method

    Hello! I need to find the rotation matrix around a given vector v=(a,b,c), by and angle ##\theta##. I can find an orthonormal basis of the plane perpendicular to v but how can I compute the matrix from this? I think I can do it by brute force, rewriting the orthonormal basis rotated by...
  14. chi_rho

    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...
  15. U

    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*...
  16. D

    A Interpretation of the EM tensor as a rotation matrix

    In special relativity, the electromagnetic field is represented by the tensor $$F^{\mu\nu} = \begin{pmatrix}0 & -E_{x} & -E_{y} & -E_{z}\\ E_{x} & 0 & -B_{z} & B_{y}\\ E_{y} & B_{z} & 0 & -B_{x}\\ E_{z} & -B_{y} & B_{x} & 0 \end{pmatrix}$$ which is an anti-symmetric matrix. Recalling the...
  17. O

    A Understanding the Connection Between Lie Algebras and Rotations

    The Lie Algebra is equipped with a bracket notation, and this bracket produces skew symmetric matrices. I know that there exists Lie Groups, one of which is SO(3). And I know that by exponentiating a skew symmetric matrix, I obtain a rotation matrix. ----------------- First, can someone edit...
  18. M

    MHB Applying rotation matrix to make inclined plane flat again

    I want to rotate an inclined plane to achieve a flat surface. I think I can use the Euler angles to perform this operation. Using following data: and following rotation matrix I think you can make the plane flat by following rotations: 1: rotation around x-axis by 45° 2: rotation around...
  19. M

    Spherical vectors and rotation of axes

    I have a velocity vector as a function of a latitude and longitude on the surface of a sphere. Let us assume I have a point V(lambda, phi) where V is the velocity. The north pole of this sphere is rotated and I have a new north pole and I have a point V'(lambda, phi) in the new system. I have...
  20. L

    Why Do Different Definitions of Rotation Matrices Exist in Mathematics?

    Happy new year. Why everybody uses this definition of rotation matrixR(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\[0.3em] \sin\theta & \cos\theta \\[0.3em] \end{bmatrix} ? This is clockwise rotation. And we always use counter clockwise in...
  21. ognik

    MHB Exploring the 3-D Rotation Matrix with Euler Rotations and Net Angle of Rotation

    The question mentions an orthogonal matrix describing a rotation in 3D ... where $\phi$ is the net angle of rotation about a fixed single axis. I know of the 3 Euler rotations, is this one of them, arbitrary, or is there a general 3-D rotation matrix in one angle? If I build one, I would start...
  22. W

    Proving the Existence of a Rotation Matrix from Given Relations

    Homework Statement Let A∈M2x2(ℝ) such that ATA = I and det(A) = -1. Prove that for ANY such matrix there exists an angle θ such that A = ## \left( \begin{array}{cc} cos(\theta) & sin(\theta)\\ sin(\theta) & -cos(\theta)\\ \end{array} \right) ## It is not sufficient to show that this matrix...
  23. brotherbobby

    Proving "Rotation Matrix is Orthogonal: Necessary & Sufficient

    I'd like to prove the fact that - since a rotation of axes is a length-preserving transformation, the rotation matrix must be orthogonal. By the way, the converse of the statement is true also. Meaning, if a transformation is orthogonal, it must be length preserving, and I have been able to...
  24. R

    Camera on Airplane: Extrinsic plus Intrinsic 3D Rotations?

    Apologies up front for the long question … I have tried to be brief. I want to define camera angles for Google Earth (GE) when rotated about an aircraft yaw axis. The input is Latitude, Longitude, Altitude plus Heading, Pitch and Bank angles, actually coming from Flight Simulator. These drive...
  25. ELB27

    Proving a certain orthogonal matrix is a rotation matrix

    Homework Statement Let ##U## be a ##2\times 2## orthogonal matrix with ##\det U = 1##. Prove that ##U## is a rotation matrix. Homework EquationsThe Attempt at a Solution Well, my strategy was to simply write the matrix as $$U = \begin{pmatrix} a & b\\ c & d \end{pmatrix}$$ and using the given...
  26. santos2015

    Rotation matrix between two orthonormal frames

    I am reading a paper and am stuck on the following snippet. Given two orthonormal frames of vectors ##(\bf n1,n2,n3)## and ##(\bf n'1,n'2,n'3)## we can form two matrices ##N= (\bf n1,n2,n3)## and ##N' =(\bf n'1,n'2,n'3)##. In the case of a rigid body, where the two frames are related via...
  27. B

    MHB Eigenvectors of 2*2 rotation matrix

    Hi Folks, I calculate the eigenvalues of \begin{bmatrix}\cos \theta& \sin \theta \\ - \sin \theta & \cos \theta \end{bmatrix} to be \lambda_1=e^{i \theta} and \lambda_2=e^{-i \theta} for \lambda_1=e^{i \theta}=\cos \theta + i \sin \theta I calculate the eigenvector via A \lambda = \lambda V as...
  28. T

    Understanding Rotation Matrices: A Journey of Mistakes and Lessons Learned

    I'm working through Meisner Thorne and Wheeler (MTW), but have been temporarily sidetracked by a problem with rotation matrices. I've worked through the maths and produced the matrices by multiplying the three individual rotation matrices, (no problem there) but I have been trying to work out...
  29. S

    Scaling the parameter of the SO(2) rotation matrix

    For the distance function ##(\Delta s)^2 = (\Delta r)^2 + (r \Delta \theta)^2##, the rotation matrix is ##R(\theta) = \begin{pmatrix} cos\ \theta & - sin\ \theta \\ sin\ \theta & cos\ \theta \end{pmatrix}##. That means that for the distance function ##(\Delta s)^2 = (\Delta r)^2 +...
  30. H

    Rotation matrix about an arbitrary axis

    Suppose a position vector v is rotated anticlockwise at an angle ##\theta## about an arbitrary axis pointing in the direction of a position vector p, what is the rotation matrix R such that Rv gives the position vector after the rotation? Suppose p = ##\begin{pmatrix}1\\1\\1\end{pmatrix}## and...
  31. A

    Rotation matrix multiplied by matrix of column vectors?

    Hey, let's say that in 2D space we have a 2x2 rotation matrix R. Normally you could multiply this rotation matrix by a 2x1 column matrix / vector X. In that case it would be XR to get the vector rotated in the way described by R. Now what I'm wondering is, what if I had 3 column vectors that I...
  32. S

    Solve Rotation Matrix Homework: 60,892857 & 12,496875 -4,837500

    Homework Statement A have two points 60,892857 12,496875 -4,837500 70,714286 14,915625 -5,240625 I have to rotate for -0,067195795 radians. Homework Equations ##\begin{bmatrix} {x}'\\ {y}' \end{bmatrix}=\begin{bmatrix} cos\alpha & -sin\alpha \\ sin\alpha & cos\alpha...
  33. Z

    Deducir la Matriz de Rotación 2D y Encontrar Ayuda

    I was trying to deduce the 2D Rotation Matrix and I got frustrated. So, I found this article: Ampliación del Sólido Rígido/ (in Spanish). I don't understand the second line. How does he separate the matrix in two different parts? Thanks for your time.
  34. C

    Show that +1 is an eigenvalue of an odd-dimensional rotation matrix.

    Homework Statement The probelm is to show, that a rotation matrix R, in a odd-dimensional vector space, leaves unchanged the vectors of at least an one-dimensional subspace. Homework Equations This reduces to proving that 1 is an eigenvalue of Rnxn if n is odd. I know that a rotational...
  35. G

    How to convert angular velocity to rotation matrix?

    Hi, I have two questions related to angular velocity: 1. According to rotational damper, Torque = Viscous Damping Coefficient * Angular Velocity. This equation gives the unit of Angular Velocity as meter square per second. How is it equivalent to rad per second? 2. If I have an angular...
  36. U

    Which Matrix Represents a Rotation About the Z-axis?

    Homework Statement Which matrix represents a rotation? Homework Equations The Attempt at a Solution It seems odd that this matrix has somewhat the form for rotation about z-axis, just that you need to swap the cos θ for the sin θ.
  37. S

    Finding a rotation matrix (difficult)

    Homework Statement There are two coordinate systems which have different euler angles. Approximately find the euler angles of the second coordinate system with respect to the first coordinate system. Do this by taking the fact that you are able to plot points and know the position of the...
  38. G

    Prove that a rotation matrix preserves distance

    Homework Statement Prove that a rotation matrix in R3 preserves distance. Such that if A is a 3*3 orthogonal rotation matrix then |A.v|=|v|. I know one can prove this is in R2 by using a trig representation of a rotation matrix and then simplifying. Is there an analogue method in R3 or some...
  39. F

    Matrix of eigenvectors, relation to rotation matrix

    So I am given B=\begin{array}{cc} 3 & 5 \\ 5 & 3 \end{array}. I find the eigenvalues and eigenvectors: 8, -2, and (1, 1), (1, -1), respectively. I am then told to form the matrix of normalised eigenvectors, S, and I do, then to find S^{-1}BS, which, with S = \frac{1}{\sqrt{2}}\begin{array}{cc} 1...
  40. Y

    Rotation matrix arbitrary axis?

    Hello everyone. I'm having some trouble with rotation matrices. I'm given three matrices J_1, J_2, J_3 which form a basis for the set of skew-symmetric matrices (\mathfrak{so}_3). Further, the matrix exponent function is such that exp[\alpha J_i]=R_i(\alpha), so taking the...
  41. M

    Find rotation matrix that diagonalizes given inertia tensor

    Homework Statement In a particular coordinate frame, the moment of inertia tensor of a rigid body is given by I = {{3,40},{4,9,0},{0,0,12}} in some units. The instantaneous angular velocity is given by ω = (2,3,4) in some units. Find a rotation matrix a that transforms to a new coordinate...
  42. N

    Rotation Matrix in 3D: Correcting Errors in 3D Coordinate System Rotation

    Homework Statement Hi I have a coordinate system as attached, and I want to rotate it along the y-axis in the clockwise direction as shown. For this purpose I use R = \left( {\begin{array}{*{20}c} {\cos \theta } & 0 & {\sin \theta } \\ 0 & 1 & 0 \\ { - \sin \theta } & 0 & {\cos...
  43. H

    Dot and Cross Product from Rotation Matrix

    I'm just learning this Latex(sic) formatting, so it's not ideal. I was trying to explore the geometrical significance of the cross product when I happened upon an interesting observation. I've seen things like this before, but never had time to really examine them. I define two vectors...
  44. R

    Converting Rotation matrix to operate on fractional coordinates

    Hi I have an orthogonalized rotation matrix -0.500000 -0.866025 0.000000 0.866025 -0.500000 0.000000 0.000000 0.000000 1.000000 for the following unit cell: a b c alpha beta gamma space group 131.760 131.760 120.910...
  45. Q

    Decomposition of a Rotation Matrix

    Hey guys! I'm new here, so forgive me if I'm posting in the wrong section. I recently picked up a book on robotics and it had a section about rotation matrices. I'm having a difficult time with the decomposition of rotation matrices. Everywhere I look, I can find the the equations for the roll...
  46. L

    Translation using rotation matrix

    Hi, I want to calculate the coordinates of an object after a particular translation. I have the 3D coordinates at the origin: x0,y0,z0 and i have the 3x3 rotation matrix: (r11, r12, r13; r21, r22, r23; r31, r32, r33) If I want to move 3 units forward, in the direction i am facing and two...
  47. H

    FEA - Rotation Matrix of Angular Deflection

    I am trying to use FEA with space frame element. I know that for rotating an angle a around the z-axis, the translational displacements of the local and global coordinates are related through the rotation matrix: \begin{bmatrix}cos(a) & sin(a) & 0 \\ -sin(a) & cos(a) & 0 \\ 0 & 0 &...
  48. D

    Deriving the Rotation Matrix of a Beam Element

    Homework Statement The rotation matrix below describes a beam element which is rotated around three axes x,y and z. Derive the rotation matrix. http://img194.imageshack.us/img194/3351/60039512.png http://img808.imageshack.us/img808/159/64794596.png Homework Equations -The Attempt at a...
  49. R

    How to compute the rotation matrix

    Hai , I have two matrix , let say X=[ A1 A2 A3 A4 A5 A5 A7 A8 A9 A10 A11 A12] and X0=[B1 B2 B3 B4 B5 B6 B7 B8 B9...
  50. L

    Eigenvalues of a rotation matrix

    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...