I recently learned how I can project a vector $\overline{a}$ onto another one $\overline{b}$. I was wondering how I could achieve the same effect, but project the vector $\overline{a}$ along a direction. For example, in this [![enter image description here](https://i.stack.imgur.com/jhQgL.png)](https://i.stack.imgur.com/jhQgL.png), we can see that the direction of the projection is the axis $\overline{b}$ normal. This I know how to do. What I really wanted to do, is that the projection was made in the direction of the vector $\overline{a}$ normal.
While searching, I found something about Thales Theorem, but I have no clue of how I could use it to my advantage. If someone could enlighten me, I would be much appreciated. 

Thank you.

EDIT: I want to get the vector $\overline{a3}$:
[![enter image description here](https://i.imgur.com/jJDibn9.png)](https://i.imgur.com/jJDibn9.png)