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

Prolog

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

Attention

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Reflection

Reflection of a vector v.

Reflection at the x-axis

Reflection at y=x Line

Rotation by 90°

B = RG

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;

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

Example:

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 and Matrices.

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