» Required Reading » Table of Contents » Chapter 3 3.2 Rotating Coordinates in a Euclidean Space
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If we rotate between components i and j by angle (so that the i direction rotates toward j) the components 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.