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3.2 Rotating Coordinates in a Euclidean Space

If we rotate basis vectors i ' ^ and j ' ^ by angle θ 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 ⁡ θ + v j sin ⁡ θ

and v j becomes

v j ' = − v i sin ⁡ θ + v j cos ⁡ θ

These effects are illustrated in the accompanying applet. You can move the vectors and also rotate the basis.

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