Matrices geometric Reflection and Shear 
Geometric operations, like rotating a position vector by a certain angle around some axis, can be achieved by multiplying the vector by an appropriate matrix. This page descript how such matrices are constructed. 


Matrix Reflection 
We will consider a point P with the coordinates (x, y) in a twodimensional space. 
In the two dimensional space we draw the vector as 


The matrix below produces a reflection of the vector across the Xaxis 
this results in the formula


The following figure shows an example with values in RedCrab Calculator 


A 3dimensional reflection of the Y position is achieved by the following formula 

To produce reflections in the X or Zplanes, place a negative sign on the
corresponding diagonal elements of the unit matrix. 

Matrix Shear 
The insertion of a nonzero element into a position outside the diagonal of the unit matrix produces a shear distortion of the position vector. 
Example: 
corresponds to


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