This site is being phased out.

Rotation

From Mathematics Is A Science
Jump to navigationJump to search
Rotation - vectors.jpg

Example. Consider a linear function with

  • $A(e_1) = e_2$;
  • $A(e_2) = -e_1$.

Then $$A(\alpha e_1 + \alpha_2e_2) = \alpha_1A(e_1) + \alpha_2A(e_2)$$ $$ = \alpha_1e_2 - \alpha_2e_1$$ $$ = -\alpha_2e_1 + \alpha_1e_2.$$ This is an example of rotation!

Rotation is a linear transformation of the plane.


Example. Rotation, rotated through $\alpha$: \[ R = \left| \begin{array}{ccc} \lambda \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array} \right|.\]

Change of variables in integrals - rotation.jpg

Then $$\det R = \cos ^2 \alpha + \sin ^2 \alpha = 1,$$ no stretching.

Exercise. What is $R$ in polar coordinates?

Theorem. All rotations are homotopic.