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If we rotate basis vectors i' and j' by angle from i and j, (so that the i' direction rotates toward j) the components of a fixed vector v change as follows:
v_{i }becomes
v_{i}' = v_{i} cos + v_{j }sin
and v_{j} becomes
v_{j}' =  v_{i} sin + v_{j} cos
These effects are illustrated in the accompanying applet. You can move the vectors and also rotate the basis.
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