Coordinates Transformation

Coordinates transformation simply involves the expression of a coordinate, originally expressed in a coordinate system (2), into a new coordinate system (1). When deriving the coordinates transformation, there is the common way which requires to know a cosine equivalence. As a handout, here's, perhaps, a more intuitive way. 
From the graph

but since: 

we can do:

That can be represented in the usual matrix form:

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