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No problem, glad it worked out so quickly!

Dot product is a very simple operation, it's just the sum of every vector component multiplied with the same component of the other vector (so the 2D case is 2 multiplications and 1 addition), so it should be faster than e.g. doing a geometric solution with trigonometry most of the time. The only thing it really does is tell you how parallel two vectors are, but it's the best choice if you need that exact thing!

Hello, I'm from Poland. I'm currently playing Heart of ruins final Columbus.  So far I have acquired 8 weapons and 48 knowledge.

 However, I don't know how to destroy the block with blue lightning.  I also don't know how to solve the puzzles for which I have hints.  I solved one with the letters "a" "b" "y" but I don't know if it's correct because they keep flashing alternately.

 I would like to ask you to help me in this game, for which I will be very grateful.  I know you've probably forgotten about this game, but I hope you'll remember something

 Regards

 My email is arko.jr@gmail.com