pseudovector

tronter

1. What constraints must the elements [tex] (R_{ij}) [/tex] of the three dimensional rotation matrix satisfy in order to preserve the length of [tex] \bold{A} [/tex] (for all vectors [math] \bold{A} [/tex]).

I am guessing that it must equal the identity matrix?


2. How do the components of a vector transform under a translation of coordinates ([tex] \bar{x} = x, \ \ \bar{y} = y-a, \ \ \bar{z} = z [/tex])? The components dont change at all (e.g. because vectors are independent of coordinate system)?


3. How do components of a vector transform under an inversion of coordinates ([tex] \bar{x} = -x, \ \ \ \bar{y} = - y, \ \ \ \bar{z} = -z [/tex])? Guessing that the components change in sign, but stay the same in magnitude?

4. How does the cross product of two vectors transform under an inversion. Is the cross product of two pseudovectors, a vector or a pseudovector?

Is it [tex] \bold{A} \times \bold{B} = - \bold{B} \times \bold{A} [/tex]? The cross product of two pseudovectors is a pseudovector? Two examples of pseudovectors in classical mechanics are torque and angular momentum?

5. How does the scalar triple product of three vectors transform under inversions (e.g. pseudoscalar).

So [tex] \bold{A} \cdot (\bold{B} \times \bold{C}) = -(\bold{B} \times \bold{C}) \cdot \bold{A} [/tex]?