Surface normal Vector

Hey!Can you please explain how you calculated them?

As in if any surface is given ,in 2D, how do I get the normal vector in (dy or dx),2D vector form.

For example if I have lets say a vector at an angle in the above image(unlike perpendicular to the surfaces). What would be the normal vector of that vector.
 
There are several things I should make clear:
1. I did not calculate these. (As the appearance suggests) this is just a version of the solution to your problem, in which the normal directions have been corrected.
It is available here: https://www.aia.rwth-aachen.de/vluebfiles/excercises/18/link/exercise1.pdf
2. I know nothing about fluid dynamics. I simply recognise that the directions of the normal vectors are wrong in the solution you have and are correct in the new version.
That is just a matter of simple vector geometry:
1615473476661.png
The unit normals to each surface are as shown, and I have simply written them in component form.

That is the extent of my knowledge, I am sorry.
However, again it would appear there is a problem with this second solution. If you look at the dot products in 3 and 4, the middle expressions do not equate to the expressions on the right. I guess that the dy and dx in the first brackets should not be there. (Again this is a guess, without knowledge of fluid dynamics).
1615473528320.png
I'm sorry I could not be of more help.
 
I attach what appears to be a reasonably good explanation of what is going on.
 

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