### Rotating Lines

**Problem:**

Given a line of slope *m* in the Euclidean plane, what is the slope *m’* of the line rotated (counterclockwise) by angle *θ*?

**Solution:**

Suppose we have an equation for the line of the form *y* = *mx* + *b*. We can ignore *b* as it is unrelated to the slope (in effect, we are working in an affine space).

So, *y* = *mx* for our purposes. Every point satisfying this equation is a multiple of

and, similarly, every point satisfying the equation *y* = *m’x* of the rotated line is a multiple of

Since the latter point is the image of the former after rotation by *θ*, the points are related by a rotation matrix, like so:

Solving for *m’* then yields

which, of course, is our solution in terms of *m* and *θ*.