# 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

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).

View More On Wikipedia.org
1. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...

26. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...