# Chapter 10: Coordinate Transformations

## Basics

Multiplying a matrix by a vector allows us to produce a new, changed vectors.

Such a transformation can be used to scale, shear, and rotate a vector or Cartesian coordinates.

## Rotation

### Matrix derivation:

There is a requisite diagram for understanding this which is coming soon…

$\vec b$ is $\vec a$ rotated by $\theta$

$\vec a$ makes angle $\alpha$ with the x-axis and its length is $r = x_a^2 + y_a^2$

Therefore:

And:

Using trigonometric relationships:

Thus:

Substituting the equations for $x_a$ and $y_a$:

And the transformation matrix is revealed.