orthogonal basis transform

Here's a trick I learnt from Strang - to convert (rotation/translation) between two arbitrary frames:

1. convert both to orthonormal bases (A,B above), ignoring the translation component for now.
   
2. use the inverse == transpose trick to find the inverse of the first transform (A^T), then apply (multiply by) the second transform(B)
3. (...fudge the translation component (not shown)...)

It's even better if you want to axis-align something, you just need the transposed matrix. This got a post of it's own because I've been told by several professors over the years to move it to the origin, calculate the x and y rotational components and multiply together - this takes lots of multiplication, and you end up with a hideous zombie-like matrix.

The transpose is equal to the inverse only if it's an orthonormal matrix - to project vector line onto another, it's a simple dot product. What's more each axis in an orthogonal frame is independent, so that's all we need - 9 dot products, or one matrix multiplication.