There are two Bloch sphere simulators that I know of available on the internet to help you visualize what happens geometrically to a qubit as a point on the surface of a Bloch sphere as well as what happens algebraically to its complex matrix elements.

The first is written in Java by Stephen Shary and Marc Cahay at the University of Cincinnati.

The second is written in Wolfram’s Computable Docment Format and requires a plugin to run.

For the purposes of this class, we won’t be covering in more detail the relationship between the geometric rotation angle and the form of the 2×2 matrix which corresponds to a single-qubit quantum gate. However, interested readers can find out more in Dave Bacon’s lecture notes, under Section 1: One Qubit. In other sources, this relationship is called the homomorphism between the groups SU(2) and SO(3), if you want to research the topic on your own, on Wikipedia or in math textbooks.