  We report a study of the Majorana geometrical representation of a qutrit,
where a pair of points on a unit sphere represents its quantum states. A
canonical form for qutrit states is presented, where every state can be
obtained from a one-parameter family of states via $SO(3)$ action. The notion
of spin-1 magnetization which is invariant under $SO(3)$ is geometrically
interpreted on the Majorana sphere. Furthermore, we describe the action of
several quantum gates in the Majorana picture and experimentally implement
these gates on a spin-1 system (an NMR qutrit) oriented in a liquid crystalline
environment. We study the dynamics of the pair of points representing a qutrit
state under various useful quantum operations and connect them to different NMR
operations. Finally, using the Gell Mann matrix picture we experimentally
implement a scheme for complete qutrit state tomography.
