RedCrab Math Tutorial

# 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 two-dimensional space.
In the two dimensional space we draw the vector as

The matrix below produces a reflection of the vector across the X-axis

this results in the formula

The following figure shows an example with values in RedCrab Calculator

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

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

## Matrix Shear

The insertion of a non-zero element into a position outside the diagonal of the unit matrix produces a shear distortion of the position vector.
Example:

corresponds to