3.2 Rotating Coordinates in a Euclidean Space

]> 3.2 Rotating Coordinates in a Euclidean Space

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3.2 Rotating Coordinates in a Euclidean Space

If we rotate basis vectors $\hat{i'}$ and $\hat{j'}$ by angle $θ$ from $\hat{i}$ and $\hat{j}$ , (so that the $\hat{i'}$ direction rotates toward $\hat{j}$ ) the components of a fixed vector $\overset{⟶}{v}$ 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.