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Vectors, forgotten math?

In the below article, can anyone please guide me in:

I) where my derivation has gone wrong, and

II) (considering the question only as the sum of two vectors, i.e. forgetting the physics part of it) why does the fallacy occur?

Please help me as the problem is quite disturbing to me.

Note by Nilabha Saha
3 weeks, 6 days ago

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