Wacker Art Rotation Matrices ## 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 and Rotation

Reflection of a vector v.

### 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

### 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 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