]> 3.2 Rotating Coordinates in a Euclidean Space

## 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.