Have you ever wondered why in a standard three phase system that the leg to leg voltage is a little lower than the two leg voltages added up? Typically, if your two legs are 120 volts to ground, the voltage between them will only be around 208 volts. A system like this uses a transformer arrangement called a wye because it looks something like a wye on paper (page 544 of Fundamentals of HVACR). The three windings are tied together in the middle, and that is where the transformer is grounded. Each leg to ground is around 120 volts, but when you check the voltage between legs, you only get 208 volts. Remember that voltage is a measurement of potential difference – it is measuring the difference in electrical potential from one point to another. That is why if you put both leads on the same point you read 0 volts. That does not mean there is no electrical potential at that point, just that the difference between your two probes is 0. When measuring the voltage between two phases of a 3 phase system, the meter is reading the difference between the two phases. If you were to draw both phases on a graph and plot another curve that represented the difference in the two phases, you would be plotting what the voltmeter reads. This can be represented mathematically by vectors.
A vector has both a direction and a strength. An easy way to understand vectors is to imagine a wagon with a rope pulling it. With one rope, the wagon will go at the same speed and direction as the person pulling the rope. Tie a second rope to the wagon and have two people pulling in different directions, and now the wagon will travel a path somewhere between the two people. Now have one person pull faster than the other, and the wagon will travel a little more in that direction. Adding different AC voltages and currents is done with vectors because vectors can account for differences in strength and direction. If the two legs were 180° apart from each other pulling in completely opposite directions, the voltage would be the sum of the two legs, as in single phase systems. Because the two phases of a wye type 3 phase system are not pulling in completely opposite directions, the difference is not simply the sum of the two voltages. You can see this by looking at a drawing of a Y type transformer, like the one on page 544 of Fundamentals of HVACR.