The 2D Rotation Matrix

Nick V. Flor • March 20, 2017 • @ProfessorF

Mathematicians on the internet are not only incomprehensible but annoying with some of their conventions: confusing subscripts, over reliance on symbols, etc.  They either don’t know how to communicate or they’re purposely trying to look smart.

Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius — and a lot of courage — to move in the opposite direction.

 – E. F. Schumacher, Economist

One thing that particularly annoys me is how they treat rotation matrices.  This is what almost every mathematician on the internet gives as a rotation matrix:

Cos(angle) -Sin(angle)
Sin(angle) Cos(angle)

The reason I don’t like this rotation matrix is because it describes a clockwise rotation, which is BACKWARDS from what EVERY STUDENT is taught in school. For example, remember how you drew a 45 degree angle in junior high? Starting at the x-axis, you swept your pencil halfway between the x and y-axis, in a counter-clockwise motion, then drew a line. Same with a 90 degree angle, you swept your pencil counter-clockwise starting from the x-axis until it was vertical.

This is the “correct” rotation matrix, where “correct” means it corresponds to how we were taught rotation in school:

Cos(angle) Sin(angle)
-Sin(angle) Cos(angle)

Why mathematicians provide a “backwards” rotation matrix is beyond me. But I’m told one should never attribute to malice that which can be explained by ignorance. So I’ll close this very short essay by pleading with Math departments to force their mathematicians to take a communications or a technical writing course as a graduation requirement.