## 3.2 Rotating Coordinates in a Euclidean Space

**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.