Reflection and Rotation
Reflection of a vector v.
Reflection and rotation of a vector v, can be expressed as a the matrix product with this vector and special matrices.
The result of this product is the reflected or rotated vector. In the following graphics the original vector is drawn in black.
The vector is drawn in green after the application of the G matrix, in red after the application of the R matrix and in blue after the
application of the B matrix.
Reflection at the xaxis
Multipying the vector v with the G matrix will result in a reflection of the original vector at the xaxis.
Example:
Reflection at y=x Line
Multipying the vector
v with the R matrix will result in a reflection of the original vector at the x=y line.
Example:
Rotation by 90° in the Counter Clockwise Direction
The combination of a reflection at the xaxis and the reflextion at the yaxis will result
in a rotation by 90 degree in the counter clockwise direction (right handed rotation).
B = RG
Example:
Two Reflections and a 90° Degree Rotation
The three example operations from above in one diagram:
Rotation by 90° in the Counter Clockwise Direction  Right Handed Rotations by 90°
Applying the righthanded rotation by the B matrix several times will result in the following rotations:
b = B^{}v 
b = B^{2}v 
b = B^{3}v 
b = B^{4}v 




Rotation by 90° in the Clockwise Direction  Left Handed Rotations by 90°
The left handed rotation or clockwise rotation by 90° is performend by the L Matrix L = B
The combination of a reflection at the yaxis and the reflextion at the xaxis will result
in a rotation by 90 degree in the clockwise direction (left handed rotation).
L = GR
Example:
Applying the lefthanded rotation by the L matrix several times will result in the following rotations:
b = L^{}v 
b = L^{2}v 
b = L^{3}v 
b = L^{4}v 



