I'm going through Classical Dynamics of Particles and Systems from Thorton and Marion, and I'm the section 1.3 is showing how to deal with coordinate transformations.
Using the drawing below, the authors state:
The new coordinate $ x'_{1} $ is the sum of the projection of $ x_1 $ on the $ x'_{1} $-axis (line $\overline{Oa}$) plus the projection of $ x_{2} $ onto the $ x'_{1} $-axis (the line $\overline{ab} $ + $\overline{bc}) $
Now, I get the first part, where, $ \overline{Oa} = x_{1} \cos{\theta} $, but when it comes to the second projection I don't know if the text correlates to the drawing completely or the authors are assuming something I'm not aware of.
If we are looking at the projection of $ x_{2} $ onto the $ x'_{1} $-axis, shouldn't that be a perpendicular line from $ x_{2} $ to the $ x'_{1} $-axis? I'm a little bit confused as the authors missed that in the drawing
In that case, how do we reason that the distance from $ O $ to that projection is equal to the distance $\overline{ac}? $
And finally, once the previous two are justified, what the intuition behind adding the two projections to get the new $ x'_{1} $ coordinate?
