Page 5 -- 2D Transformations Using 2x2 Matrices

A transformation matrix can be used to define rotation, translation, stretch, or the projection of a vector or set of vectors in a defined coordinate system. This page is intended to provide the user with an opportunity to compare the mathematical manipulations with the resulting physical process.

PURE ROTATION

To define a vector with respect to a coordinate system which has been rotated about a line perpendicular to the axis plane

In two dimensions (or the XY plane), the rotation necessarily occurs about the unseen z-axis. The resultant vector position is a projection of the new orientation angle, and therefore is a function of SINE and COSINE:

For example, a unit vector in the x-direction, rotated 90 degrees counterclockwise would result aligned in the y-direction, as evidenced below:

ROTATION DEMO

PURE TRANSLATION

To change the endpoint of a vector while maintaining orientation

TRANSLATION DEMO

STRETCH

To change the magnitude of a vector

STRETCH DEMO