Q:

How to find a "least squares" line with a known slope?

I have gps trackings that I know they fall into a certain pattern - a line with a known angle. How do I find the line that minimizes the distances of the points from it but is in the correct angle?
Unfortunately, I can't post an image for example.

A:

If I well understand, you have a set of $n$ points $(x_1,y_1),(x_2,y_2),...,(x_k,y_k),..., (x_n,y_n)$ and you want to fit a straight line $y=ax+b$ which coefficient $a$ is known.
This is a regression problem of the kind  $Y(x)=b$ where $Y(x)=y(x)-ax$
Hense $$b\simeq \frac{1}{n}\sum_{k=1}^n (y_k-ax_k)$$
In this particular case ($a$ known), the adjusted value of $b$ is the same if we consider the orthogonal distances. 

