Wacker Art Rotation Matrices Wappen der Familie Wacker
Beach Travemuende
Bild: "Beach"


This page is based on the concepts developed on my geometric algebra page and on my linear algebra page.


This page has been tested and developed with the Mozilla browser. This page requires JavaScript in your browser. Without JavaScript or JavaScript beeing disabled, the display of this side is incomplete.

Reflection and Rotation

Reflection of a vector v.

Reflection at the x-axis


Reflection at y=x Line


Rotation by 90°

The combination of a reflection at the x-axis and the reflextion at the y-axis will result in a rotation by 90 degree.

B = RG


Two Reflections and a 90° Degree Rotation

The three example operations from above in one diagram:

Group Properties

The following four matrices form a group.

   B2 = -I;    R2 = I;     G2 = I

   B = RG;    R = BG;    G = RB;

  -B = GR;   -R = GB;   -G = BR;

The Relation to the geometric algebra G2 can be found on the geometric algebra and matrices page:

Relation to Complex Numbers

Complex numbers z can be represented by the following matrix equation: z = xI + yB.

Rotation Matrices

If R is a n×n matrix and the value of the determinant of R; det(R) = 1; then R is a rotation matrix.

Multiplying the rotation matrix R with the vector v which has n elements will result in a vector v'.

v' = Rv

A rotation matrix R change only the direction of a vector v when this vector is multiplied with the matrix R and leave the length of a the vector v unchanged:

|v'| = |Rv| = |v|

2×2 Rotation Matrices

Let the rotation matrix have the following form:

Rotation Examples

Rotation Formular


Application of Different Rotation Matrices on Vectors

Unit vectors e1 and e2:

Application of the unit vectors e1 and e2 on the general rotation matrix R(θ):

Application of different rotation matrices on the vectors e1 and e2:

Rotations in Three Dimensions

Rotation Around the x-Axis

Rotation Around the y-Axis

Rotation Around the z-axis

The next page is about Geometric Algebra.

18. Dezember 2017 Version 1.0
Copyright: Hermann Wacker Uhlandstraße 10 D-85386 Eching bei Freising Germany Haftungsausschluß