How does a bicycle remain stable when in motion? I could understand it at a theoretical standpoint but how can one represent the dynamic stability of the bicycle mathematically using a vector diagram ?
 A: As you know, a moving bicycle stays upright because the rider turns the steering wheel "into the fall". The interesting thing about a well-designed bicycle (and I hint at this in my answer about the curved fork of the bicycle is that when the bike tilts out of the vertical plane, a torque is generated that will turn the front wheel into the fall. This is why it is possible to ride a bike without holding on to the handlebars, and why it is stable.
Since you asked for a vector diagram, here it is (the first is borrowed from my earlier answer):

On the left is a "bicycle", showing the bent fork. Note the red line - it indicates the "tail" - the distance between the axis of steering, and the contact point between wheel and ground.
In the middle is a view of the front wheel as it tilts, showing that there will be a torque on the wheel
On the right is a view of the front wheel along the steering tube: now you can see that the reaction force of the road causes the wheel turns to the right when the bike tilts to the right.
And when the wheel turns to the right, there will be a restoring torque that keeps the bike from falling over...
