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Rotation Matrices

Rotation matrices are square matrices used to mathematically represent the rotation of vectors and coordinate systems by a specific angle around a given axis. They are a fundamental tool in many fields including linear algebra, geometry, physics, engineering, and computer graphics.

Mathematical Foundation

In an n-dimensional vector space, the rotation of an object or coordinate system is a linear transformation. This transformation can be represented by a matrix. Rotation matrices are special square matrices that represent such linear rotations.

2D Rotation

In two-dimensional Euclidean space (R2), the counterclockwise rotation of a vector or coordinate system about the origin by an angle θ is represented by the following rotation matrix:

R(θ)=(cosθsinθ−sinθcosθ)

The rotated version of a vector v=(xy) by an angle θ is denoted v′ and is computed as follows:

v′=R(θ)v=(cosθsinθ−sinθcosθ)(xy)=(xcosθ−ysinθxsinθ+ycosθ)

3D Rotation

In three-dimensional Euclidean space (R3), rotations are defined by an axis and an angle. The counterclockwise rotation matrices about the principal axes are as follows:

Rotation by angle α about the x-axis: Rx(α)=<svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="3.600em" viewBox="0 0 875 3600"><path d="M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1 c-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349, -36,557 l0,84c0.2,6,0,26,0,60c2,159.3,10,310.7,24,454c53.3,528,210, 949.7,470,1265c4.7,6,9.7,11.7,15,17c0.7,0.7,7,1,19,1c0,0,18,0,18,0c4,-4,6,-7,6,-9 c0,-2.7,-3.3,-8.7,-10,-18c-135.3,-192.7,-235.5,-414.3,-300.5,-665c-65,-250.7,-102.5, -544.7,-112.5,-882c-2,-104,-3,-167,-3,-189 l0,-92c0,-162.7,5.7,-314,17,-454c20.7,-272,63.7,-513,129,-723c65.3, -210,155.3,-396.3,270,-559c6.7,-9.3,10,-15.3,10,-18z"/></svg>1000cosαsinα0−sinαcosα<svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="3.600em" viewBox="0 0 875 3600"><path d="M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3, 63,95c96.7,156.7,172.8,332.5,228.5,527.5c55.7,195,92.8,416.5,111.5,664.5 c11.3,139.3,17,290.7,17,454c0,28,1.7,43,3.3,45l0,9 c-3,4,-3.3,16.7,-3.3,38c0,162,-5.7,313.7,-17,455c-18.7,248,-55.8,469.3,-111.5,664 c-55.7,194.7,-131.8,370.3,-228.5,527c-20.7,34.7,-41.7,66.3,-63,95c-2,3.3,-4,7,-6,11 c0,7.3,5.7,11,17,11c0,0,11,0,11,0c9.3,0,14.3,-0.3,15,-1c5.3,-5.3,10.3,-11,15,-17 c242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558 l0,-144c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7, -470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z"/></svg>

Rotation by angle β about the y-axis: Ry(β)=<svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="3.600em" viewBox="0 0 875 3600"><path d="M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1 c-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349, -36,557 l0,84c0.2,6,0,26,0,60c2,159.3,10,310.7,24,454c53.3,528,210, 949.7,470,1265c4.7,6,9.7,11.7,15,17c0.7,0.7,7,1,19,1c0,0,18,0,18,0c4,-4,6,-7,6,-9 c0,-2.7,-3.3,-8.7,-10,-18c-135.3,-192.7,-235.5,-414.3,-300.5,-665c-65,-250.7,-102.5, -544.7,-112.5,-882c-2,-104,-3,-167,-3,-189 l0,-92c0,-162.7,5.7,-314,17,-454c20.7,-272,63.7,-513,129,-723c65.3, -210,155.3,-396.3,270,-559c6.7,-9.3,10,-15.3,10,-18z"/></svg>cosβ0−sinβ010sinβ0cosβ<svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="3.600em" viewBox="0 0 875 3600"><path d="M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3, 63,95c96.7,156.7,172.8,332.5,228.5,527.5c55.7,195,92.8,416.5,111.5,664.5 c11.3,139.3,17,290.7,17,454c0,28,1.7,43,3.3,45l0,9 c-3,4,-3.3,16.7,-3.3,38c0,162,-5.7,313.7,-17,455c-18.7,248,-55.8,469.3,-111.5,664 c-55.7,194.7,-131.8,370.3,-228.5,527c-20.7,34.7,-41.7,66.3,-63,95c-2,3.3,-4,7,-6,11 c0,7.3,5.7,11,17,11c0,0,11,0,11,0c9.3,0,14.3,-0.3,15,-1c5.3,-5.3,10.3,-11,15,-17 c242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558 l0,-144c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7, -470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z"/></svg>

Rotation by angle γ about the z-axis: Rz(γ)=<svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="3.600em" viewBox="0 0 875 3600"><path d="M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1 c-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349, -36,557 l0,84c0.2,6,0,26,0,60c2,159.3,10,310.7,24,454c53.3,528,210, 949.7,470,1265c4.7,6,9.7,11.7,15,17c0.7,0.7,7,1,19,1c0,0,18,0,18,0c4,-4,6,-7,6,-9 c0,-2.7,-3.3,-8.7,-10,-18c-135.3,-192.7,-235.5,-414.3,-300.5,-665c-65,-250.7,-102.5, -544.7,-112.5,-882c-2,-104,-3,-167,-3,-189 l0,-92c0,-162.7,5.7,-314,17,-454c20.7,-272,63.7,-513,129,-723c65.3, -210,155.3,-396.3,270,-559c6.7,-9.3,10,-15.3,10,-18z"/></svg>cosγsinγ0−sinγcosγ0001<svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="3.600em" viewBox="0 0 875 3600"><path d="M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3, 63,95c96.7,156.7,172.8,332.5,228.5,527.5c55.7,195,92.8,416.5,111.5,664.5 c11.3,139.3,17,290.7,17,454c0,28,1.7,43,3.3,45l0,9 c-3,4,-3.3,16.7,-3.3,38c0,162,-5.7,313.7,-17,455c-18.7,248,-55.8,469.3,-111.5,664 c-55.7,194.7,-131.8,370.3,-228.5,527c-20.7,34.7,-41.7,66.3,-63,95c-2,3.3,-4,7,-6,11 c0,7.3,5.7,11,17,11c0,0,11,0,11,0c9.3,0,14.3,-0.3,15,-1c5.3,-5.3,10.3,-11,15,-17 c242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558 l0,-144c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7, -470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z"/></svg>

In three dimensions, a general rotation can be obtained by sequentially applying these basic rotations. For example, complex rotations can be defined using Euler angles (yaw, pitch, roll) or angle-axis representations, and their corresponding rotation matrices are found as products of these basic matrices. However, it is important to note that the order of matrix multiplication matters (R1R2=R2R1 generally holds).

Properties of Rotation MatricesR1R2=R2R1

Orthogonal Matrices: Rotation matrices are orthogonal, meaning their transpose (RT) equals their inverse (R−1):

RT=R−1⟹RRT=RTR=I

Here, I denotes the identity matrix. This property implies that rotations preserve lengths and angles.

Determinant is +1: The determinant of a rotation matrix is always +1:

det(R)=+1

A determinant of +1 indicates that the transformation preserves orientation (e.g., the right-hand rule). Orthogonal matrices with determinant −1 represent reflections.

Applications

Rotation matrices are used to perform fundamental transformations across various disciplines:

Geometry and Linear Algebra: Used to rotate points, vectors, and geometric shapes around a specified axis. They enable the transformation of coordinate systems into different orientations.

Physics: Used to describe the orientation and transformations of objects in rigid body mechanics, rotational motion, and angular momentum. For example, rotation matrices are essential for analyzing the motion of a gyroscope or a spinning top.

Engineering: Used to model and control the orientation of objects and sensors in robotics, control systems, and aerospace. They are vital for computing joint rotations in robotic arms or determining the attitude of an aircraft in space.

Computer Graphics: A fundamental tool for rotating objects on screen in 3D modeling, animation, and game development. They are heavily used to control camera movements and object orientations in virtual space.

Computer Vision: Used to analyze and match different views of objects in image processing and object recognition tasks.

Navigation and Cartography: Used to compute transformations between global coordinate systems and provide directional information.

Author Information

AuthorAlper CangüdenDecember 9, 2025 at 7:48 AM

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Mathematical Foundation
- 2D Rotation
- 3D Rotation
Properties of Rotation Matrices
Applications