Here is a good exercise in vector identities and matrices, leading to some important general results: $($a$)$ For a rigid body made up of particles of mass m$,$ spinning about an axis through the origin with angular velocity $,$ prove that its total kinetic energy can be written as

$$

m$[($r$)2($$$r$)2].$ Remember that v $=$ $$ r$.$ You may find the following vector identity useful: For any two vectors

a and b$,$

$($a$$b$)2=$a$2$b$2($a$$b$)2.($If you use the identity, please prove it$.)($b$)$ Prove that the angular momentum L of the body can be

written a